circle page Parts of a Circle Circle - The set of all points in a plane that is the same distance from a given point called the center. Arc - Part of a circle; the curved line between two points on a circle. If the circle is divided into two unequal parts, the smaller arc is called the minor arc and the larger arc is called the major arc. Center (of a Circle) - The point inside the circle that is the same distance from all points on the circle. Closed Curve - A curve that can be traced with the same starting and stopping points, and without crossing or retracing any part of the curve. A circle is an example of a closed curve. EG: You may want to have a separate diagram illustrating interior, on, and exterior regions of the circle. The circle is the set of points that lie on the blue curve. The points that lie inside this curve are not part of the circle (this is called the disk). A circle is an example of a closed curve because you can begin at one point on the circle and trace it, ending up where you started without crossing or retracing any part of the circle.

If the circle is divided into two unequal parts or arcs, the smaller arc is called the minor arc and the larger arc is called the major arc. EG: I would talk about how your would name a circle and its arcs. The center of a circle is the same distance from all points on the circle. Point C is the center of this circle. This distance from the center of the circle to any point on the circumference is called the radius (indicated by r in the picture at the right). Any line segment whose endpoints are the center and a point on the circle is also called the radius. Point E is not the center of this circle. Point E is not even part of the circle. Based on this definition, "same distance from all points", can a square have a center? Measurements Related to a Circle EG: Any reason why the formula for the area of a circle is not written as *r2? I would rephase the definition of diameter as the segment passing through the center containing endpoints on the circle. In the definition of pi, I would use the approximate symbol and mention that it has an infinite number of decimal places.

In the definition of radius, I would replace the word "circumference" with "circle." Area of a Circle - The amount of surface within a given boundary. Area is measured in squared units. The area of a circle can be found using A = pi *r*r (* means to multiply). Circumference - The distance around a circle. The circumference can be found using C = 2*pi*r (* means to multiply). Diameter - The distance across a circle through its center. The line segment that includes the center and whose endpoints lie on the circumference of the circle. Pi - The ratio of the circumference of a circle to its diameter ( =3.1416). Radius - The distance from the center of a circle to any point on the circumference. The line segment that has the center of the circle as one endpoint and a point on the circle as the other endpoint. The circumference (indicated in red) is like the perimeter around the circle, indicating the length or distance around the circle. (It's similar, but not exactly alike.

Perimeter deals with straight lines while circumference deals with curved lines.) As the number of sides of an inscribed regular polygon increases (see figures below), the perimeter of the regular polygon gets closer to the circumference of the circle. In other words, the perimeter of the regular octagon more closely approximates the circumference of the circle than does the perimeter of the regular pentagon or the equilateral triangle. The ratio of the circumference of any circle to its diameter is always a number that is close to 3.1416. This number is called pi and is written . Pi is an irrational number, which means it cannot be written as an exact decimal. We usually estimate pi with 3.14, 3.1416, or 22/7. EG: Again, I would change the word "circumference" to "circle" (or just omit the words "the circumference of"). The line segment that goes through the center of the circle and whose endpoints are on the circumference of the circle is called a diameter. The length of this line segment is also called the diameter. EG: Again, I would omit the words "the circumference of".

The line segment that has one endpoint as the center of the circle and the other endpoint on the circumference of the circle is called the radius of the circle. What do you think the relationship is between the radius and the diameter? If you aren't sure, measure the diameter and radius of a circle. What do you notice? Chord - A line segment that joins two points on the circumference of circle Examples of Chords The far right example shows the diameter as a chord of the circle. So the length of a chord is equal to or less than the diameter. Non-Examples of Chords All Elementary Mathematics - Study Guide - Geometry - Geometrical locus. Circle and circumference... Geometrical locus. Circle and circumference Geometrical locus. Midperpendicular. Angle bisector. Circumference. Circle. Arc. Secant. Chord. Diameter. Tangent line. Segment of a circle. Sector of a circle. Angles in a circle. Central angle. Inscribed angle. Circumscribed angle. Radian measure of angles. Round angle. Ratio of circumference length and diameter. Length of an arc. Huygens' formula. Relations between elements of a circle.

Geometrical locus ( or simply locus ) is a totality of all points, satisfying the certain given conditions. E x a m p l e 1. A midperpendicular of any segment is a locus, i.e. a totality of all points, equally removed from the bounds of the segment. Suppose that PO AB and AO = OB : Then, distances from any point P, lying on the midperpendicular PO, to bounds A and B of the segment AB are both equal to d . So, each point of a midperpendicular has the following property: it is removed from the bounds of the segment at equal distances. E x a m p l e 2. An angle bisector is a locus, that is a totality of all points, equally removed from the angle sides. E x a m p l e 3. A circumference is a locus, that is a totality of all points ( one of them - A ), equally removed from its center O. Circumference is a geometrical locus in a plane, that is a totality of all points, equally removed from its center.

Each of the equal segments, joining the center with any point of a circumference is called a radius and signed as r or R . A part of a plane inside of a circumference, is called a circle. A part of a circumference ( for instance, AmB, Fig.39 ) is called an arc of a circle. The straight line PQ, going through two points M and N of a circumference, is called a secant ( or transversal ). Its segment MN, lying inside of the circumference, is called a chord. A chord, going through a center of a circle ( for instance, BC, Fig.39 ), is called a diameter and signed as d or D . A diameter is the greatest chord of a circle and equal to two radii ( d = 2 r ). Tangent. Assume, that the secant PQ ( Fig.40 ) is going through points K and M of a circumference. Assume also, that point M is moving along the circumference, approaching the point K. Then the secant PQ will change its position, rotating around the point K. As approaching the point M to the point K, the secant PQ tends to some limit position AB.

The straight line AB is called a tangent line or simply a tangent to the circumference in the point K. The point K is called a point of tangency. A tangent line and a circumference have only one common point пїЅ a point of tangency. Properties of tangent. is an irrational number; its approximate value is 3.1415926пїЅ On the other hand, 2 is a round angle of a circumference, which in a degree measure is equal to 360 deg. In practice it often occurs, that both radius and angle of a circle are unknown. In this case, an arc length can be calculated by the approximate HuygensпїЅ formula: p 2l + ( 2l пїЅ L ) / 3 , where ( according to Fig.42 ): p пїЅ a length of the arc ACB; l пїЅ a length of the chord AC; L пїЅ a length of the chord AB. If an arc contains not more than 60 deg, a relative error of this formula is less than 0.5%. Relations between elements of a circle.

An inscribed angle ( ABC, Fig.45 ) is equal to a half of the central angle ( AOC, Fig.45 ), based on the same arc AmC. Therefore, all inscribed angles ( Fig.45 ), based on the same arc ( AmC, Fig.45 ), are equal. As a central angle contains the same quantity of degrees, as its arc ( AmC, Fig.45 ), then any inscribed angle is measured by a half of an arc, which is based on ( AmC in our case ). All inscribed angles, based on a semi-circle ( APB, AQB, пїЅ, Fig.46 ), are right angles ( Prove this, please ! ). An angle ( AOD, Fig.47 ), formed by two chords ( AB and CD ), is measured by a semi-sum of arcs, concluded between its sides: ( AnD + CmB ) / 2 . An angle ( AOD, Fig.48 ), formed by two secants ( AO and OD ), is measured by a semi-difference of arcs, concluded between its sides: ( AnD пїЅ BmC ) / 2 .

An angle ( DCB, Fig.49 ), formed by a tangent and a chord ( AB and CD ), is measured by a half of an arc, concluded inside of it: CmD / 2 .An angle ( BOC, Fig.50 ), formed by a tangent and a secant ( CO and BO ), is measured by a semi-difference of arcs, concluded between its sides: ( BmC пїЅ CnD ) / 2 . A circumscribed angle ( AOC, Fig.50 ), formed by the two tangents, (CO and AO), is measured by a semi-difference of arcs, concluded between its sides: ( ABC пїЅ CDA ) / 2 . Products of segments of chords ( AB and CD, Fig.51 or Fig.52 ), into which they are divided by an intersection point, are equal: AO пїЅ BO = CO пїЅ DO. A square of tangent line segment is equal to a product of a secant line segment by the secant line external part ( Fig.50 ): OA2 = OB пїЅ OD ( prove, please! ). This property may be considered as a particular case of Fig.52.

A chord ( AB, Fig.53 ), which is perpendicular to a diameter ( CD ), is divided into two in the intersection point O : AO = OB . ( Try to prove this ! ). Circles and using a Compass Year 8 Interactive Maths - Second Edition Circles and using a Compass In this section, we will consider the definition of a circle, lines in a circle and parts of a circle. Circle A circle is a set of points in a plane that are the same distance from a fixed point (called the centre). These set of points form the perimeter of the circle. The radius is the distance from the centre of the circle to any point on its perimeter. The circumference of a circle is the perimeter of the circle. These parts of a circle are indicated in the accompanying diagram. Note: The plural of radius is radii. Lines in a Circle The name of a line in a circle depends on its position in the circle. A secant is a line that passes through any two points on a circle. A chord is a line that joins two points on the circumference of a circle. The diameter is a chord that passes through the centre of a circle. A tangent is a line that touches the circle at only one point.

Parts of a Circle An arc is a part of the circumference . A sector is the part of a circle between two radii . A segment is the part of a circle that is between a chord and the circumference . A semicircle is a half of a circle. Compass A compass is an instrument used to draw circles or the parts of circles called arcs . It consists of two movable arms hinged together where one arm has a pointed end and the other arm holds a pencil. Note that a compass is also called a pair of compasses. To draw a circle (or arc) with a compass: make sure that the hinge at the top of the compass is tightened so that it does not slip tighten the hold for the pencil so it also does not slip align the pencil lead with the compass's needle press down the needle and turn the knob at the top of the compass to draw a circle (or arc ) Example 2 Use a compass to draw a circle of radius 4 cm. Solution: Step 1: Use a ruler to set the distance from the point of the compass to the pencil's lead at 4 cm. Step 2: Place the point of the compass at the centre of the circle.

Step 3: Draw the circle by turning the compass through 360ï¿½. Activity 10.1 1. Use a compass to draw a circle of radius 5 cm. 2. Use a compass to draw a circle of diameter 12 cm. 3a. Use a compass to draw a circle of radius 4.5 cm. 3b. Draw the diameter of the circle; and use a ruler to measure the length of the diameter. 3c. Write an equation to represent the relation between the radius, r, and the diameter, d. 4a. Use a compass to draw a circle of radius 5.5 cm. 4b. Draw a diameter and label it PQ. 4c. Draw a triangle PQR where R is on the semicircle. 4d. Use a protractor to measure the size of angle PRQ. 5a. Use a compass to draw a circle of radius 6.5 cm. 5b. Draw a diameter and label it PQ. 5c. Draw a triangle PQR where R is on the semicircle. 5d. Use a protractor to measure the size of angle PRQ. 6a.

Use a compass to draw a circle of radius 7.5 cm. 6b. Draw a diameter and label it PQ. 6c. Draw a triangle PQR where R is on the semicircle. 6d. Use a protractor to measure the size of angle PRQ. 7. Use the results of questions 4, 5 and 6 to complete the following statements: a. The size of the angle on the diameter of a circle with a vertex on the circle is ï¿½ b. If a triangle is drawn in a semicircle using the diameter as an edge, the angle touching the curved part of the triangle is ï¿½ Key Terms What Is Circle? What Is Circle? Circle is the locus of points equidistant from a given point, the center of the circle. The common distance from the center of the circle to its points is called radius. Thus a circle is completely defined by its center (O) and radius (R): C(O, R) = O(R) = {x: dist(O, x) = R}. This definition that appears so natural to a 21st century reasonably educated person is markedly different from the one given by Euclid (Definition I.15).

In Sir Heath's translation: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another. And in the translation by Robert Simson : A circle is a plane figure contained by one line, which is called circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. As an extra, comes Definition I.16: And the point is called the center of the circle. What is so different between our definition and Euclid's? Strange as it may seem, Euclid did not have a notion of metric or a distance function. Nowhere in the Elements he refers to a distance between two points, but only to the length of the line segment joining the two. The whole of geometry has been developed without the idea of a 2d-distance! And so it does not seem to be necessary. The definition we gave which is the most common nowadays is the outgrowth of the culture permeated by analytic geometry. Ironically, in the modern terminology being Euclidean exactly means being in possession of a suitable metric. Euclid might have objected.

Now, returning to the 21st century, our definition is meaningful in any metric space . For some metrics the circles may look rather square . However, a circle looks like a circle in Euclidean geometry . Although the Euclidean circle has much to recommend it for, long, long ago Piet Hein has discovered that circles in the metric dist((x1, y1), (x2, y2)) = ((x1 - x2)5/2 + (y1 - y2)5/2)2/5 are more pleasant to the eye than the Euclidean ones. For square circles, it is relatively simple to define their length as the sum of lengths of the constituent sides. The length of the Euclidean (ordinary) circle may be approximately measured to any precision with a variety of means. A rigorous definition, nonetheless, requires a theory of limits - the fundamental notion in Calculus . The length of a circle is usually called its circumference and is equal 2πR, where R is the radius of the circle. The area enclosed by the circle can also be computed with a simple formula: πR². In any affine space endowed with a metric, all circles are similar.

Which does not mean there is similarity between circles in various metric spaces. But which does mean that, however defined, the ratio of the circumference to the radius of circles in a given metric space is constant. It may vary between spaces, though. In Euclidean space, the ratio equals 2p, named so by L. Euler. The most common tool for constructing (Euclidean) circles is the venerable compass. But there are of course other ways. The inversion can be used to convert circular motion into rectilinear and vice versa. A circle is also traced by the point common to two lighthouse beams rotating at the same angular speed. The midpoint of a ladder sliding down a wall also traces a circle. Circle is also the locus of points having the same ratio of distances to a pair of points . Circle (a part of it actually) is also the locus of points from which a given line segment is seen under a given angle. The line joining the center of the circle to one of its points is often called the radius-vector of that point, or just one of the radii. Likewise, the word diameter is used in two different, but close meanings. On one hand, diameter is the largest distance between any two points on a circle.

As such, it equals twice the radius of the circle. For a given circle, there is exactly one and only one diameter. On the other hand, the word "diameter" designates a line segment joining a point on the circle with its opposite, i.e. the point farthest away from the point at hand. For every point on a circle there is exactly one diameter that contains this point. The length of a diameter, as a line segment, is exactly the diameter, as a number. Any diameter passes through the center of the circle and is divided by the center into two equal halves, each the length of the radius. In Cartesian coordinates , circle is defined by a second degree equation: (x - x0)2 + (y - y0)2 = R2, where (x0, y0) is the center of the circle and R its radius. The polar coordinates of a circle centered at the origin are particularly simple: r = R. A segment of a line joining two points on a circle is called a chord; a piece of a circle between two points is an arc. A chord that passes through the center of the circle is (ambiguously) called a diameter. Diameter is the longest chord in a circle .

A shape bounded by an arc and a chord with the same end points is a segment. A central angle is formed by two radius-vectors. A central angle cuts from a circle a sector. A straight line that has only one point in common with a circle is tangent to the circle. Other than tangents, straight lines cross a circle either in no points or in two points. The lines of the latter variety are called secants. Two secants that meet on a circle define an inscribed angle . Two radii define a central angle . Two intersecting chord are divided by the point of intersection into two parts each such that the products of their lengths are equal. This is known as the Intersecting Chords Theorem . The latter carries on to the intersecting secants and gives rise to the important notion of the power of a point with respect to a circle . Triangles always have one inscribed and one circumscribed circle. Three more circles (excircles) touch the triangle's sidelines. Many other special circles are associated with a triangle: 9-Point circle , Taylor circle , Tucker circles , Adams' circle , to name a few. Archimedes has discovered his Twin Circles in the wonderful arbelos .

Some quadrilaterals have inscribed circles, and these are called inscriptible . Other quadrilaterals have concyclic vertices, and these are called circumscriptible , or cyclic. Those quadrilaterals that have both inscribed and circumscribed circles are said to be bicentric . Surprisingly, in many respects circles behave like straight lines. Nowehere it is more apparent than in inversive geometry . An Aside The famous French mathematician Alexandre Grothendieck - a 1966 Fields medalist - has an uncommonly tragic life story [ Ruelle ]: growing up in a concentration camp and later being an outsider in the French mathematics establishment. He was one of the first permanent professors at the Institut des Hautes Études Scientifiques, a position which he resigned in protest of the institute being partially funded by military grants. Years later he applied for a position at Le Centre National de la Recherche Scientifique. As an application, he submitted Esquisse d'un Programme ( Sketch of a Program , in English translation), where he summarized his past activity and outlined his plans for future research.

He described one of his results, namely (p 15 in the original , p 246 in the translation ), Every "finite" oriented map is canonically realised on a complex algebraic curve. A few lines later he wrote, This discovery, which is technically so simple, made a very strong impression on me, and it represents a decisive turning point in the course of my reflections, a shift in particular of my centre of interest in mathematics, which suddenly found itself strongly focused. I do not believe that a mathematical fact has ever struck me quite so strongly as this one, nor had a comparable psychological impact. This was accompanied with a foot note (p 44 in the original , p 274 in the translation ): With the exception of another "fact", at the time when, around the age of twelve, I was interned in the concentration camp of Rieucros (near Mende). It is there that I learnt, from another prisoner, Maria, who gave me free private lessons, the definition of the circle. It impressed me by its simplicity and its evidence, whereas the property of "perfect rotundity" of the circle previously had appeared to me as a reality mysterious beyond words.

It is at that moment, I believe, that I glimpsed for the first time (without of course formulating it to myself in these terms) the creative power of a "good" mathematical definition, of a formulation which describes the essence. Still today, it seems that the fascination which this power exercised on me has lost nothing of its force. (My sincere thanks go to V. Gutenmacher who brought the quote to my attention and to Ethan Duckworth for suggesting a more complete reference.) Circles Circles When you have finished this page, try the Circles Quiz . A circle is a plane figure with all points on the outside or circumference equally distant from the center of the figure. A line that is drawn from a point on one side of the circle to a point on the other side and passes through the center is the diameter of the circle. The diameter is a line segment, and thus is named by the letters that are at each endpoint. A line segment drawn from the center of the circle to a outside point of the circle is called a radius (plural radii). The radius is named by the letters at its endpoints. A chord is a line segment that touches two points on the outside of the circle but doesn't pass through the center.

The measure of the distance around the outside of a circle or the circle's perimeter is its circumference. A tangent is a line or line segment that touches the outside of the circle at a point, but does not pass through it in any way. A secant is a line that goes through or intersects a circle at two points and extends beyond the limits of the circle. An arc is a part of a circumference of the circle that has two endpoints on the circumference of the circle. Geometry and the Circle Unit 2 > Lesson 1 of 6 A circle is an important shape in the field of geometry. Let's look at the definition of a circle and its parts. We will also examine the relationship between the circle and the plane. A circle is a shape with all points the same distance from its center. A circle is named by its center. Thus, the circle to the right is called circle A since its center is at point A. Some real world examples of a circle are a wheel, a dinner plate and (the surface of) a coin. The distance across a circle through the center is called the diameter. A real-world example of diameter is a 9-inch plate.

The radius of a circle is the distance from the center of a circle to any point on the circle. If you place two radii end-to-end in a circle, you would have the same length as one diameter. Thus, the diameter of a circle is twice as long as the radius. We can look at a pizza pie to find real-world examples of diameter and radius. Look at the pizza to the right which has been sliced into 8 equal parts through its center. A radius is formed by making a straight cut from the center to a point on the circle. A straight cut made from a point on the circle, continuing through its center to another point on the circle, is a diameter. As you can see, a circle has many different radii and diameters, each passing through its center. A chord is a line segment that joins two points on a curve. In geometry, a chord is often used to describe a line segment joining two endpoints that lie on a circle. The circle to the right contains chord AB. If this circle was a pizza pie, you could cut off a piece of pizza along chord AB. By cutting along chord AB, you are cutting off a segment of pizza that includes this chord. A circle has many different chords.

Some chords pass through the center and some do not. A chord that passes through the center is called a diameter. It turns out that a diameter of a circle is the longest chord of that circle since it passes through the center. A diameter satisfies the definition of a chord, however, a chord is not necessarily a diameter. This is because every diameter passes through the center of a circle, but some chords do not pass through the center. Thus, it can be stated, every diameter is a chord, but not every chord is a diameter. Let's revisit the definition of a circle. A circle is the set of points that are equidistant from a special point in the plane. The special point is the center. In the circle to the right, the center is point A. Thus we have circle A. A plane is a flat surface that extends without end in all directions. In the diagram to the right, Plane P contains points A, B and C. Can you think of some real world objects that satisfy the definition of a plane? At this level of mathematics, that is difficult to do. Intuitively, a plane may be visualized as a flat infinite sheet of paper.

The top of your desk and a chalkboard are objects which can be used to represent a plane, although they do not satisfy the definition above. A circle divides the plane into three parts: the points INSIDE the circle the points OUTSIDE the circle and the points ON the circle the points inside the circle the points outside the circle the points on the circle You can see an interactive demonstration of this by placing your mouse over the three items below. A circle divides a plane into three parts: the points INSIDE the circle the points OUTSIDE the circle and the points ON the circle Example 1: The length of DB is 2.5 inches Summary: A circle is a shape with all points the same distance from its center. A circle is named by its center. The parts of a circle include a radius, diameter and a chord. All diameters are chords, but not all chords are diameters. A plane is a flat surface that extends without end in all directions. A circle divides the plane into three parts: The points inside the circle, the points outside the circle and the points on the circle. Exercises Directions: Refer to the diagram to answer each question below. Select your answer by clicking on its button.

Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button. 1. Which of the following is a chord, but not a diameter? Circle Circle A circle is easy to make: Draw a curve that is "radius" away from a central point. All points are the same distance from the center. You Can Draw It Yourself Put a pin in a board, put a loop of string around it, and insert a pencil into the loop. Keep the string stretched and draw the circle! Play With It Try dragging the point to see how the radius and cirfcumference change. Radius, Diameter and Circumference The Radius is the distance from the center outwards. The Diameter goes straight across the circle, through the center. The Circumference is the distance once around the circle. And here is the really cool thing: When we divide the circumference by the diameter we get 3.141592654... which is the number π ( Pi ) So when the diameter is 1, the circumference is 3.141592654...

We can say: Circumference = π × Diameter Example: You walk around a circle which has a diameter of 100m, how far have you walked? Distance walked = Circumference = π × 100m = 314m (to the nearest m) Also note that the Diameter is twice the Radius: Diameter = 2 × Radius And so this is also true: Circumference = 2 × π × Radius The length of the words may help you remember: Radius is the shortest word Diameter is longer (and is 2 × Radius) Circumference is the longest (and is π × Diameter) Definition A = 3.14159... × 1.44 = 4.52 (to 2 decimals) Area Compared to a Square A circle has about 80% of the area of a similar-width square. The actual value is (π/4) = 0.785398... = 78.5398...% Names Because people have studied circles for thousands of years special names have come about.

Nobody wants to say "that line that starts at one side of the circle, goes through the center and ends on the other side" when a word like "Diameter" will do. So here are the most common special names: Lines A line that goes from one point to another on the circle's circumference is called a Chord. If that line passes through the center it is called a Diameter. A line that "just touches" the circle as it passes by is called a Tangent. And a part of the circumference is called an Arc. Slices There are two main "slices" of a circle. The "pizza" slice is called a Sector . And the slice made by a chord is called a Segment . Common Sectors The Quadrant and Semicircle are two special types of Sector: Quarter of a circle is called a Quadrant. Half a circle is called a Semicircle. Inside and Outside A circle has an inside and an outside (of course!). But it also has an "on", because we could be right on the circle. Example: "A" is outside the circle, "B" is inside the circle and "C" is on the circle. Everything Maths and Science Proof Draw \(OA\) and \(OB\).

In \(\triangle OPA\) and in \(\triangle OPB\), \[\begin{array}{rll} OA^2 &= OP^2 + AP^2 & \text{(Pythagoras)} \\ OB^2 &= OP^2 + BP^2 & \text{(Pythagoras)} \end{array}\] and \[\begin{array}{rll} OA &= OB & \text{(equal radii)} \\ \therefore AP^2 &= BP^2 & \\ \therefore AP &= BP & \end{array}\] Therefore \(OP\) bisects \(AB\). Alternative proof: In \(\triangle OPA\) and in \(\triangle OPB\), \[\begin{array}{rll} O\hat{P}A &= O\hat{P}B & (\text{given } OP \perp AB) \\ OA &= OB & \text{(equal radii)} \\ OP &= OP & \text{(common side)} \\ \therefore \triangle OPA & \equiv \triangle OPB & \text{(RHS)} \\ \therefore AP &= PB & \end{array}\] Therefore \(OP\) bisects \(AB\).

(PROOF NOT FOR EXAMS) Converse Line from circle centre to mid-point of chord is perpendicular If a line is drawn from the centre of a circle to the mid-point of a chord, then the line is perpendicular to the chord. (Reason: line from centre to mid-point \(\perp\)) Circle with centre \(O\) and line \(OP\) to mid-point \(P\) on chord \(AB\). \(OP \perp AB\) Proof Draw \(OA\) and \(OB\). In \(\triangle OPA\) and in \(\triangle OPB\), \[\begin{array}{rll} OA &= OB & \text{(equal radii)} \\ AP &= PB & \text{(given)} \\ OP &= OP & \text{(common side)} \\ \therefore \triangle OPA & \equiv \triangle OPB & \text{(SSS)} \\ \therefore O\hat{P}A &= O\hat{P}B & \\ \text{and } O\hat{P}A + O\hat{P}B &= \textrm{180}\text{°} & (\angle \text{ on str.

line}) \\ \therefore O\hat{P}A = O\hat{P}B &= \textrm{90}\text{°} & \end{array}\] Therefore \(OP \perp AB\). Perpendicular bisector of chord passes through circle centre If the perpendicular bisector of a chord is drawn, then the line will pass through the centre of the circle. (Reason: \(\perp\) bisector through centre) Circle with mid-point \(P\) on chord \(AB\). Line \(QP\) is drawn such that \(Q\hat{P}A = Q\hat{P}B = \textrm{90}\text{°}\). Line \(RP\) is drawn such that \(R\hat{P}A = R\hat{P}B = \textrm{90}\text{°}\). Circle centre \(O\) lies on the line \(PR\) Proof Use your results to make a conjecture about the relationship between angles subtended by an arc at the centre of a circle and angles at the circumference of a circle. Now draw three of your own similar diagrams and measure the angles to check your conjecture.

Angle at the centre of a circle is twice the size of the angle at the circumference If an arc subtends an angle at the centre of a circle and at the circumference, then the angle at the centre is twice the size of the angle at the circumference. (Reason: \(\angle \text{ at centre } = 2 \angle \text{ at circum.}\)) Circle with centre \(O\), arc \(AB\) subtending \(A\hat{O}B\) at the centre of the circle, and \(A\hat{P}B\) at the circumference.

Table 3 Show me the answers Practise more questions like this Answer 1: \[\begin{array}{rll} b &= 2 \times \textrm{45}\text{°} & (\angle \text{ at centre } = 2 \angle \text{ at circum.}) \\ \therefore b &= \textrm{90}\text{°} & \end{array}\] \[\begin{array}{rll} c &= \frac{1}{2} \times \textrm{45}\text{°} & (\angle \text{ at centre } = 2 \angle \text{ at circum.}) \\ \therefore c &= \textrm{22,5}\text{°} & \end{array}\] \[\begin{array}{rll} d &= 2 \times \textrm{100}\text{°} & (\angle \text{ at centre } = 2 \angle \text{ at circum.}) \\ \therefore d &= \textrm{200}\text{°} & \end{array}\] \[\begin{array}{rll} e &=

 \textrm{100}\text{°} - \textrm{90}\text{°} - \textrm{35}\text{°} & (\angle \text{ in semi circle} ) \\ \therefore e &= \textrm{55}\text{°} & \end{array}\] \[\begin{array}{rll} f &= \frac{1}{2} \times \textrm{240}\text{°} & (\angle \text{ at centre } = 2 \angle \text{ at circum.}) \\ \therefore f &= \textrm{120}\text{°} & \end{array}\] Activity 2: Investigation: Subtended angles in the same segment of a circle Measure angles \(a\), \(b\), \(c\), \(d\) and \(e\) in the diagram below: Choose any two points on the circumference of the circle and label them \(A\) and \(B\).

Draw \(AP\) and \(BP\), and measure \(A\hat{P}B\). Draw \(AQ\) and \(BQ\), and measure \(A\hat{Q}B\). What do you observe? Make a conjecture about these types of angles. Subtended angles in the same segment of a circle are equal If the angles subtended by a chord of the circle are on the same side of the chord, then the angles are equal. (Reason: \(\angle\)s in same seg.) Circle with centre \(O\), and points \(P\) and \(Q\) on the circumference of the circle. Arc \(AB\) subtends \(A\hat{P}B\) and \(A\hat{Q}B\) in the same segment of the circle. \(A\hat{P}B = A\hat{Q}B\) Proof Equal arcs subtend equal angles From the theorem above we can deduce that if angles at the circumference of a circle are subtended by arcs of equal length, then the angles are equal.

In the figure below, notice that if we were to move the two chords with equal length closer to each other, until they overlap, we would have the same situation as with the theorem above. This shows that the angles subtended by arcs of equal length are also equal. (PROOF NOT FOR EXAMS) Converse Concyclic points If a line segment subtends equal angles at two other points on the same side of the line segment, then these four points are concyclic (lie on a circle). Line segment \(AB\) subtending equal angles at points \(P\) and \(Q\) on the same side of the line segment \(AB\). \(A\), \(B\), \(P\) and \(Q\) lie on a circle. Proof Proof by contradiction: Points on the circumference of a circle: we know that there are only two possible options regarding a given point — it either lies on circumference or it does not. We will assume that point \(P\) does not lie on the circumference. We draw a circle that cuts \(AP\) at \(R\) and passes through \(A\), \(B\) and \(Q\).

\[\begin{array}{rll} A\hat{Q}B &= A\hat{R}B & (\angle \text{s in same seg.}) \\ \text{but } A\hat{Q}B &= A\hat{P}B & (\text{given}) \\ \therefore A\hat{R}B &= A\hat{P}B & \\ \text{but } A\hat{R}B &= A\hat{P}B + R\hat{B}P & (\text{ext. } \angle \triangle = \text{sum int. opp.}) \\ \therefore R\hat{B}P &= \textrm{0}\text{°} & \end{array}\] Therefore the assumption that the circle does not pass through \(P\) must be false. We can conclude that \(A\), \(B\), \(Q\) and \(P\) lie on a circle (\(A\), \(B\), \(Q\) and \(P\) are concyclic).

Example 3: Concyclic points Question Given \(FH \parallel EI\) and \(E\hat{I}F = \textrm{15}\text{°}\), determine the value of \(b\).

Show me the answers Practise more questions like this Answer 3: \[\begin{array}{rll} \text{In } \triangle WTZ &\text{ and in } \triangle YTZ, \\ WT &= YT & (\text{ given }) \\ ZT &= ZT & (\text{ common side }) \\ Y\hat{T}Z = W\hat{T}Z &= \textrm{90}\text{°} & (\text{ line from circle centre to mid-point }) \\ \therefore T\hat{Z}Y &= T\hat{Z}W & (\text{ SAS }) \\ T\hat{Z}Y = T\hat{Z}W &= f & \\ \text{And } T\hat{Z}Y &= \textrm{35}\text{°} & (\angle \text{s in same seg.}) \\ \therefore T\hat{Z}W = f &= \textrm{35}\text{°} & \end{array}\] Cyclic quadrilaterals Cyclic quadrilaterals are quadrilaterals with all four vertices lying on the circumference of a circle (concyclic).

Activity 3: Investigation: Cyclic quadrilaterals Consider the diagrams given below: Table 4 Use your results to make a conjecture about the relationship between angles of cyclic quadrilaterals. Opposite angles of a cyclic quadrilateral The opposite angles of a cyclic quadrilateral are supplementary. (Reason: opp. \(\angle\)s cyclic quad.) Circle with centre \(O\) with points \(A, B, P\) and \(Q\) on the circumference such that \(ABPQ\) is a cyclic quadrilateral. \(A\hat{B}P + A\hat{Q}P = \textrm{180}\text{°}\) and \(Q\hat{A}B + Q\hat{P}B = \textrm{180}\text{°}\) Proof Draw \(AO\) and \(OP\). Label \(\hat{O}_1\) and \(\hat{O}_2\).

\[\begin{array}{rll} \hat{O}_1 &= 2A\hat{B}P & (\angle\text{ at centre} = 2\angle \text{ at circum.})\\ \hat{O}_2 &= 2A\hat{Q}P & (\angle\text{ at centre} = 2\angle \text{ at circum.})\\ \text{and } \hat{O}_1 + \hat{O}_2 &= \textrm{360}\text{°} & (\angle\text{s around a point}) \\ \therefore 2A\hat{B}P + 2A\hat{Q}P &= \textrm{360}\text{°} & \\ A\hat{B}P + A\hat{Q}P &= \textrm{180}\text{°} & \end{array}\] Similarly, we can show that \(Q\hat{A}B + Q\hat{P}B = \textrm{180}\text{°}\).

Converse: interior opposite angles of a quadrilateral If the interior opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. Exterior angle of a cyclic quadrilateral If a quadrilateral is cyclic, then the exterior angle is equal to the interior opposite angle. Example 4: Opposite angles of a cyclic quadrilateral Question Given the circle with centre \(O\) and cyclic quadrilateral \(PQRS\). \(SQ\) is drawn and \(S\hat{P}Q = \textrm{34}\text{°}\). Determine the values of \(a\), \(b\) and \(c\). Use theorems and the given information to find all equal angles on the diagram Solve for \(b\) \[\begin{array}{rll} S\hat{P}Q + c &= \textrm{180}\text{°} & (\text{opp.

} \angle \text{s cyclic quad supp.}) \\ \therefore c &= \textrm{180}\text{°} - \textrm{34}\text{°} & \\ &= \textrm{146}\text{°} & \end{array}\] \[\begin{array}{rll} a &= \textrm{90}\text{°} & (\angle \text{ in semi circle}) \end{array}\] In \(\triangle PSQ\): \[\begin{array}{rll} a + b + \textrm{34}\text{°} &= \textrm{180}\text{°} & (\angle \text{ sum of } \triangle) \\ \therefore b &= \textrm{180}\text{°} - \textrm{90}\text{°} - \textrm{34}\text{°} & \\ &= \textrm{56}\text{°} & \end{array}\] Methods for proving a quadrilateral is cyclic There are three ways to prove that a quadrilateral is a cyclic quadrilateral: Table

 6 Show me the answers Practise more questions like this Answer 2: \[\begin{array}{rll} A\hat{M}B &= \textrm{32}\text{°} + D\hat{B}C & (\text{ext angle of } \triangle = \text{sum int.

opp. angles}) \\ \therefore \textrm{72}\text{°} &= \textrm{32}\text{°} + D\hat{B}C & \\ \therefore D\hat{B}C &= \textrm{40}\text{°} & (\text{angles sum of } \triangle) \\ \therefore D\hat{B}C &= D\hat{A}C & \\ \text{Therefore } ABCD &= \text{ is cyclic quad. } & ( \text{ angles in same seg.}) \end{array}\] \[\begin{array}{rll} \text{In } \triangle ABD, \quad A\hat{B}D = A\hat{D}B &= \textrm{35}\text{°} & (\angle \text{s opp.

equal sides }) \\ \therefore \textrm{35}\text{°} + \textrm{35}\text{°} + D\hat{A}B &= \textrm{180}\text{°} & (\angle \text{s sum of } \triangle) \\ \therefore D\hat{A}B &= \textrm{110}\text{°} & (\text{angles sum of } \triangle) \\ \text{And } D\hat{A}B + D\hat{C}B &= \textrm{180}\text{°} & \\ \text{Therefore } ABCD &= \text{ is cyclic quad. } & ( \text{ opp. int. angles supp.}) \end{array}\] Tangent line to a circle A tangent is a line that touches the circumference of a circle at only one place. The radius of a circle is perpendicular to the tangent at the point of contact. Two tangents drawn from the same point outside a circle If two tangents are drawn from the same point outside a circle, then they are equal in length.

(Reason: tangents from same point equal) Circle with centre \(O\) and tangents \(PA\) and \(PB\), where \(A\) and \(B\) are the respective points of contact for the two lines. \(AP = BP\) Proof In \(\triangle AOP\) and \(\triangle BOP\), \[\begin{array}{rll} O\hat{A}P = O\hat{B}P &= \textrm{90}\text{°} & (\text{tangent} \perp \text{radius})\\ AO &= BO & (\text{equal radii}) \\ OP &= OP & (\text{common side}) \\ \therefore \triangle AOP &\equiv \triangle BOP & (\text{RHS}) \\ \therefore AP &= BP & \end{array}\] Example 6: Tangents from the same point outside a circle Question In the diagram below \(AE = \textrm{5}\text{ cm}\), \(AC = \textrm{8}\text{ cm}\) and \(CE = \textrm{9}\text{ cm}\).

Determine the values of \(a\), \(b\) and \(c\). \(\hat{D} =\) \(\hat{E} =\) Use your results to complete the following: the angle between a tangent to a circle and a chord is \(\ldots \ldots\) to the angle in the alternate segment. Tangent-chord theorem The angle between a tangent to a circle and a chord drawn at the point of contact, is equal to the angle which the chord subtends in the alternate segment. (Reason: tan. chord theorem) Circle with centre \(O\) and tangent \(SR\) touching the circle at \(B\). Chord \(AB\) subtends \(\hat{P}_1\) and \(\hat{Q}_1\). \(A\hat{B}R = A\hat{P}B\) \(A\hat{B}S = A\hat{Q}B\) Proof Draw diameter \(BT\) and join \(T\) to \(A\). Let \(A\hat{T}B = T_1\).

\[\begin{array}{rll} A\hat{B}S + A\hat{B}T &= \textrm{90}\text{°} & (\text{tangent} \perp \text{radius}) \\ B\hat{A}T &= \textrm{90}\text{°} & (\angle \text{ in semi circle}) \\ \therefore A\hat{B}T + T_1 &= \textrm{90}\text{°} & (\angle \text{ sum of } \triangle BAT) \\ \therefore A\hat{B}S &= T_1 & \\ \text{but } Q_1 &= T_1 & (\angle \text{s in same segment}) \\ \therefore Q_1 &= A\hat{B}S & \end{array}\] \[\begin{array}{rll} A\hat{B}S + A\hat{B}R &= \textrm{180}\text{°} & (\angle \text{s on str.

line}) \\ \hat{Q}_1 + \hat{P}_1 &= \textrm{180}\text{°} & (\text{opp. } \angle \text{s cyclic quad. supp.}) \\ \therefore A\hat{B}S + A\hat{B}R &= Q_1 + P_1 & \\ \text{and } A\hat{B}S &= Q_1 & \\ \therefore A\hat{B}R &= P_1 & \end{array}\] Example 7: Tangent-chord theorem Show me the answers Practise more questions like this Answer 3: \[\begin{array}{rll} \hat{A}_1 &= A\hat{C}B & (\text{alt. angles}, AP \parallel BC) \\ A\hat{C}B &= A\hat{B}C & (\angle \text{s opp.

equal sides }, AB = AC) \\ \therefore \hat{A}_1 &= A\hat{B}C & \\ \text{Therefore } PAL & \text{is a tangent to circle } ABC & (\angle\text{ between line chord} = \angle\text{ in alt. seg.}) \end{array}\] \[\begin{array}{rll} \hat{A}_2 &= \hat{B}_2 & (\text{ given }) \\ \text{And } A\hat{P}B &= \hat{B}_2 & (\text{alt. angles}, AP \parallel BC) \\ \text{Therefore } A\hat{P}B &= \hat{A}_2 ABC & \\ \text{Therefore } AB & \text{is a tangent to circle } ADP & (\angle\text{ between line chord} = \angle\text{ in alt. seg.}) \end{array}\] Converse: tangent-chord theorem If a line drawn through the end point of a chord forms an angle equal to the angle subtended by the chord in the alternate segment, then the line is a tangent to the circle.

(Reason: \(\angle\) between line and chord \(= \angle\) in alt. seg. ) Example 8: Applying the theorems Question Diameter - math word definition - Math Open Reference From Greek: dia- "across, through" + metron "a measure" The distance across a circle through its center point. Try this Drag the orange dot. The blue line will always remain a diameter of the circle. The diameter of a circle is the length of the line through the center and touching two points on its edge. In the figure above, drag the orange dots around and see that the diameter never changes. Sometimes the word 'diameter' is used to refer to the line itself. In that sense you may see "draw a diameter of the circle". In the more recent sense, it is the length of the line, and so is referred to as "the diameter of the circle is 3.4 centimeters" The diameter is also a chord . A chord is a line that joins any two points on a circle. A diameter is a chord that runs through the center point of the circle. It is the longest possible chord of any circle. The center of a circle is the midpoint of its diameter.

That is, it divides it into two equal parts, each of which is a radius of the circle. The radius is half the diameter. If you know the radius Given the radius of a circle, the diameter can be calculated using the formula where: R is the radius of the circle If you know the circumference If you know the circumference of a circle, the diameter can be found using the formula where: C is the circumference of the circle π is Pi , approximately 3.142 If you know the area If you know the area of a circle, the diameter can be found using the formula where: π is Pi , approximately 3.142 Calculator

Question: What name is given to a straight line that joins any two points on the circumference of a circle?
Answer: Chord