Angle Relationships (Vertex On, Inside & Outside) Name Theorem Hypothesis Conclusion Tangent Chord Theorem (Vertex On) If a tangent and a chord intersect at a point on the circle, then the measure of each angle formed is one half the measure of its intercepted arc. Example: Find the measure of 1. Example: Find the measure of arc EFG. Name Theorem Hypothesis Conclusion Interior Angles of a Circle Theorem (Vertex Inside) If two chords intersect inside the circle, then the measure of each angle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Example: Find x and y. Example: Find the value of x. Name Theorem Hypothesis Conclusion Exterior Angles of a Circle Theorem (Vertex Outside) If a tangent and a secant, two tangents, or two secants intersect outside the circle, then the measure of the angle formed is half the difference of the measures of the intercepted arcs. Example: Find the value of x. Example: Find the value of x.
Name Theorem Hypothesis Conclusion Perpendicular Tangent Theorem If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Converse of Perpendicular Tangent Theorem If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. Example: Is AB tangent to Circle C? Example: Find ST. Example: Find AB. Name Theorem Hypothesis Conclusion Tangent Segments Theorem If two segments are tangent to a circle from the same external point, then the segments are congruent. Example: Find perimeter of triangle ABC. Example: Find DF if you know that DF and DE are tangent to. Chord Properties Name Theorem Hypothesis Conclusion
Converting Radians and Degrees One unit of measurement for angles is degrees, which are based on a fraction of a circle. Another unit is called radian, which is defined as the measure of a central angle whose arc length is the same as the radius of the circle. In the following picture, let r represent the length of the radius of the circle, θ represent the measure of the central angle in radians, and s represent the length of the intercepted arc. By the definition of radians, r and s are equal to each other. But what if I wanted to find the number of radians in the entire circle? I would have to take the circumference divided by the radius. To convert from radians to degrees: To convert from degrees to radians: Example: Convert from radians to degrees: Example: Convert from degrees to radians 135 270 90
D. Find the volume of a cone with base circumference 25 in. and a height 2 in. more than twice the radius. Application of Volume 1. A standard size sheet of paper measures 8.5 inches by 11 inches. Use two standard sized sheets of paper to create two cylinders. One cylinder should have a height of 11 inches and the other cylinder should have a height of 8.5 inches. Predict which cylinder will have the greatest volume: _______________________________ a. Determine the radius of each cylinder: h = 11 inches h = 8.5 inches b. Calculate the volume of each cylinder. H = 11 inches h = 8.5 inches c. Does the radius or height have a greater impact on the magnitude of the volume? Why? d. What does the radius need to be on the cylinder with a height of 11 inches so the volume of the cylinder with a height of 11 inches equals the volume of the cylinder with a height of 8 inches?