Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: In the diagram below, triangle $ABC$ is inscribed in the circle and $AC = AB$. The measure of angle $BAC$ is 42 degrees and segment $ED$ is tangent to the circle at point $C$. What is the measure of angle $ACD$? <image1>
Answer: 69{degrees}
Problem: <image>Question: $ABCDEFGH$ is a regular octagon of side 12cm. Find the area in square centimeters of trapezoid $BCDE$. Express your answer in simplest radical form.


<image1>
Answer: 72+72\sqrt{2}
Problem: <image>Question: The truncated right circular cone below has a large base radius 8 cm and a small base radius of 4 cm. The height of the truncated cone is 6 cm.  The volume of this solid is $n \pi$ cubic cm, where $n$ is an integer.  What is $n$? <image1>
Answer: 224
Problem: <image>Question: In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively.  Determine the area of quadrilateral $DBEF$. <image1>
Answer: 8
Problem: <image>Question: In the figure, $BA = AD = DC$ and point $D$ is on segment $BC$. The measure of angle $ACD$ is 22.5 degrees. What is the measure of angle $ABC$? <image1>
Answer: 45
Problem: <image>Question: The trapezoid shown has a height of length $12\text{ cm},$ a base of length $16\text{ cm},$ and an area of $162\text{ cm}^2.$ What is the perimeter of the trapezoid? <image1>
Answer: 52
Problem: <image>Question: A square is divided, as shown. What fraction of the area of the square is shaded? Express your answer as a fraction. <image1>
Answer: \frac{3}{16}
Problem: <image>Question: A paper cone is to be made from a three-quarter circle having radius 4 inches (shaded). What is the length of the arc on the discarded quarter-circle (dotted portion)? Express your answer in terms of $\pi$.

<image1>
Answer: 2\pi
Problem: <image>Question: If the point $(3,4)$ is reflected in the $x$-axis, what are the coordinates of its image?

<image1>
Answer: (3,-4)
Problem: <image>Question: The area of the semicircle in Figure A is half the area of the circle in Figure B. The area of a square inscribed in the semicircle, as shown, is what fraction of the area of a square inscribed in the circle?

<image1>
Answer: \frac{2}{5}
Problem: <image>Question: The vertices of a triangle are the points of intersection of the line $y = -x-1$, the line $x=2$, and $y = \frac{1}{5}x+\frac{13}{5}$. Find an equation of the circle passing through all three vertices.

<image1>
Answer: 13
Problem: <image>Question: Quadrilateral $QABO$ is constructed as shown. Determine the area of $QABO$. <image1>
Answer: 84
Problem: <image>Question: In the figure shown, a perpendicular segment is drawn from B in rectangle ABCD to meet diagonal AC at point X.  Side AB is 6 cm and diagonal AC is 10 cm.  How many centimeters away is point X from the midpoint M of the diagonal AC?  Express your answer as a decimal to the nearest tenth.

<image1>
Answer: 1.4
Problem: <image>Question: Let $ABCD$ be a rectangle.  Let $E$ and $F$ be points on $BC$ and $CD$, respectively, so that the areas of triangles $ABE$, $ADF$, and $CEF$ are 8, 5, and 9, respectively.  Find the area of rectangle $ABCD$.

<image1>
Answer: 40
Problem: <image>Question: In the figure below, $ABDC,$ $EFHG,$ and $ASHY$ are all squares; $AB=EF =1$ and $AY=5$.

What is the area of quadrilateral $DYES$?

<image1>
Answer: 15
Problem: <image>Question: What is the area in square inches of the pentagon shown?

<image1>
Answer: 144
Problem: <image>Question: A quarter-circle of radius 3 units is drawn at each of the vertices of a square with sides of 6 units. <image1> The area of the shaded region can be expressed in the form $a-b\pi$ square units, where $a$ and $b$ are both integers. What is the value of $a+b?$
Answer: 45
Problem: <image>Question: For triangle $ABC$, points $D$ and $E$ are   the midpoints of sides $AB$ and $AC$,  respectively.  Side $BC$ measures six inches.   What is the measure of segment $DE$ in inches?

<image1>
Answer: 3
Problem: <image>Question: The solid shown was formed by cutting a right circular cylinder in half. If the base has a radius of 6 cm and the height is 10 cm, what is the total surface area, in terms of $\pi$, of the solid? <image1>
Answer: 96\pi+120
Problem: <image>Question: A square and an equilateral triangle have	equal	perimeters.	The area of the triangle is $16\sqrt{3}$ square centimeters. How long, in centimeters, is a diagonal of the square? Express your answer in simplest radical form.

<image1>
Answer: 6\sqrt{2}
Problem: <image>Question: In the diagram, the centre of the circle is $O.$ The area of the shaded region is $20\%$ of the area of the circle. What is the value of $x?$ <image1>
Answer: 72
Problem: <image>Question: Triangle $PAB$ and square $ABCD$ are in perpendicular planes. Given that $PA=3$, $PB=4$, and $AB=5$, what is $PD$? <image1>
Answer: \sqrt{34}
Problem: <image>Question: Squares $ABCD$ and $EFGH$ are equal in area.  Vertices $B$, $E$, $C$, and $H$ lie on the same line.  Diagonal $AC$ is extended to $J$, the midpoint of $GH$.  What is the fraction of the two squares that is shaded? <image1>
Answer: \frac{5}{16}
Problem: <image>Question: If $a$, $b$, and $c$ are consecutive integers, find the area of the shaded region in the square below: <image1>
Answer: 24
Problem: <image>Question: A company makes a six-sided hollow aluminum container in the shape of a rectangular prism as shown. The container is $10^{''}$ by $10^{''}$ by $12^{''}$. Aluminum costs $\$0.05$ per square inch. What is the cost, in dollars, of the aluminum used to make one container?

<image1>
Answer: 34
Problem: <image>Question: In the figure, $ABCD$ and $BEFG$ are squares, and $BCE$ is an equilateral triangle. What is the number of degrees in angle $GCE$?

<image1>
Answer: 45
Problem: <image>Question: The vertices of a convex pentagon are $(-1, -1), (-3, 4), (1, 7), (6, 5)$ and $(3, -1)$. What is the area of the pentagon? <image1>
Answer: 47
Problem: <image>Question: In the figure below, side $AE$ of rectangle $ABDE$ is parallel to the $x$-axis, and side $BD$ contains the point $C$. The vertices of triangle $ACE$ are $A(1, 1)$, $C(3, 3)$ and $E(4, 1)$. What is the ratio of the area of triangle $ACE$ to the area of rectangle $ABDE$?

<image1>
Answer: \frac{1}{2}
Problem: <image>Question: In the figure, point $O$ is the center of the circle, the measure of angle $RTB$ is 28 degrees, and the measure of angle $ROB$ is three times the measure of angle $SOT$. What is the measure of minor arc $RS$, in degrees? <image1>
Answer: 68
Problem: <image>Question: In the diagram, points $X$, $Y$ and $Z$ are on the sides of $\triangle UVW$, as shown.  Line segments $UY$, $VZ$ and $WX$ intersect at $P$.  Point $Y$ is on $VW$ such that $VY:YW=4:3$. If $\triangle PYW$ has an area of 30 and $\triangle PZW$ has an area of 35, determine the area of $\triangle UXP$. <image1>
Answer: 84
Problem: <image>Question: In the figure shown, $AC=13$ and $DC=2$ units.  What is the length of the segment $BD$? Express your answer in simplest radical form.

<image1>
Answer: \sqrt{22}
Problem: <image>Question: Coplanar squares $ABGH$ and $BCDF$ are adjacent, with $CD = 10$ units and $AH = 5$ units. Point $E$ is on segments $AD$ and $GB$. What is the area of triangle $ABE$, in square units?

<image1>
Answer: \frac{25}{3}
Problem: <image>Question: In circle $O$, $\overline{PN}$ and $\overline{GA}$ are diameters and m$\angle GOP=78^\circ$. How many degrees are in the measure of $\angle NGA$? <image1>
Answer: 39
Problem: <image>Question: In right triangle $XYZ$, shown below, what is $\sin{X}$?

<image1>
Answer: \frac{3}{5}
Problem: <image>Question: The right pyramid shown has a square base and all eight of its edges are the same length. What is the degree measure of angle $ABD$?<image1>
Answer: 45
Problem: <image>Question: A rectangular box is 4 cm thick, and its square bases measure 16 cm by 16 cm. What is the distance, in centimeters, from the center point $P$ of one square base to corner $Q$ of the opposite base? Express your answer in simplest terms.

<image1>
Answer: 12
Problem: <image>Question: In $\triangle{RST}$, shown, $\sin{R}=\frac{2}{5}$.  What is $\sin{T}$?

<image1>
Answer: \frac{\sqrt{21}}{5}
Problem: <image>Question: The lines $y = -2x + 8$ and $y = \frac{1}{2} x - 2$ meet at $(4,0),$ as shown. What is the area of the triangle formed by these two lines and the line $x = -2?$ <image1>
Answer: 45
Problem: <image>Question: In the diagram, $PRT$ and $QRS$ are straight lines. What is the value of $x$? <image1>
Answer: 55
Problem: <image>Question: In the triangle, $\angle A=\angle B$. What is $x$? <image1>
Answer: 3
Problem: <image>Question: Four circles of radius 1 are each tangent to two sides of a square and externally tangent to a circle of radius 2, as shown.  What is the area of the square?

<image1>
Answer: 22+12\sqrt{2}
Problem: <image>Question: In the diagram, square $ABCD$ has sides of length 4, and $\triangle ABE$ is equilateral. Line segments $BE$ and $AC$ intersect at $P$. Point $Q$ is on $BC$ so that $PQ$ is perpendicular  to $BC$ and $PQ=x$. <image1>

Find the value of $x$ in simplest radical form.
Answer: 2\sqrt{3}-2
Problem: <image>Question: The following diagonal is drawn in a regular heptagon, creating a pentagon and a quadrilateral.  What is the measure of $x$, in degrees? 

<image1>
Answer: \frac{360}7
Problem: <image>Question: A semicircle is constructed along each side of a right triangle with legs 6 inches and 8 inches. The semicircle placed along the hypotenuse is shaded, as shown. What is the total area of the two non-shaded crescent-shaped regions? Express your answer in simplest form.

<image1>
Answer: 24
Problem: <image>Question: A unit circle has its center at $(5,0)$ and a second circle with a radius of $2$ units has its center at $(11,0)$ as shown. A common internal tangent to the circles intersects the $x$-axis at $Q(a,0)$. What is the value of $a$? <image1>
Answer: 7
Problem: <image>Question: In the diagram, $K$, $O$ and $M$ are the centers of the three semi-circles. Also, $OC = 32$ and $CB = 36$. <image1> What is the area of the shaded region?
Answer: 900\pi
Problem: <image>Question: In triangle $ABC$, $AB = 13$, $AC = 15$, and $BC = 14$.  Let $I$ be the incenter.  The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively.  Find the area of quadrilateral $AEIF$.

<image1>
Answer: 28
Problem: <image>Question: What is the angle of rotation in degrees about point $C$ that maps the darker figure to the lighter image? <image1>
Answer: 180
Problem: <image>Question: $ABCDEFGH$ shown below is a right rectangular prism.  If the volume of pyramid $ABCH$ is 20, then what is the volume of $ABCDEFGH$?

<image1>
Answer: 120
Problem: <image>Question: Segment $AB$ measures 4 cm and is a diameter of circle $P$. In triangle $ABC$, point $C$ is on circle $P$ and $BC = 2$ cm. What is the area of the shaded region?

<image1>
Answer: 4\pi-2\sqrt{3}
Problem: <image>Question: What is the number of square centimeters in the shaded area? (The 10 represents the hypotenuse of the white triangle only.) <image1>
Answer: 30
Problem: <image>Question: Four semi-circles are shown with $AB:BC:CD = 1:2:3$. What is the ratio of the shaded area to the unshaded area in the semi circle with diameter $AD$? <image1>
Answer: \frac{11}{7}
Problem: <image>Question: Rectangle $WXYZ$ is drawn on $\triangle ABC$, such that point $W$ lies on segment $AB$, point $X$ lies on segment $AC$, and points $Y$ and $Z$ lies on segment $BC$, as shown. If $m\angle BWZ=26^{\circ}$ and $m\angle CXY=64^{\circ}$, what is $m\angle BAC$, in degrees? <image1>
Answer: 90
Problem: <image>Question: $ABCD$ is a square with $AB = 8$cm. Arcs $BC$ and $CD$ are semicircles. Express the area of the shaded region, in square centimeters, and in terms of $\pi$. (As always, do not include units in your submitted answer.) <image1>
Answer: 8\pi-16
Problem: <image>Question: A decorative arrangement of floor tiles forms concentric circles, as shown in the figure to the right. The smallest circle has a radius of 2 feet, and each successive circle has a radius 2 feet longer. All the lines shown intersect at the center and form 12 congruent central angles. What is the area of the shaded region? Express your answer in terms of $\pi$. <image1>
Answer: \pi
Problem: <image>Question: Given that $\overline{MN}\parallel\overline{AB}$, how many units long is $\overline{BN}$?

<image1>
Answer: 4
Problem: <image>Question: All of the triangles in the figure and the central hexagon are equilateral. Given that $\overline{AC}$ is 3 units long, how many square units, expressed in simplest radical form, are in the area of the entire star? <image1>
Answer: 3\sqrt{3}
Problem: <image>Question: The lateral surface area of the frustum of a solid right cone is the product of one-half the slant height ($L$) and the sum of the circumferences of the two circular faces. What is the number of square centimeters in the total surface area of the frustum shown here? Express your answer in terms of $\pi$.

<image1>
Answer: 256\pi
Problem: <image>Question: What is the area in square units of the quadrilateral XYZW shown below? <image1>
Answer: 2304
Problem: <image>Question: A hexagon is inscribed in a circle: <image1> What is the measure of $\alpha$, in degrees?
Answer: 145
Problem: <image>Question: By joining alternate vertices of a regular hexagon with edges $4$ inches long, two equilateral triangles are formed, as shown. What is the area, in square inches, of the region that is common to the two triangles? Express your answer in simplest radical form. <image1>
Answer: 8\sqrt{3}{squareinches}
Problem: <image>Question: A greeting card is 6 inches wide and 8 inches tall. Point A is 3 inches from the fold, as shown. As the card is opened to an angle of 45 degrees, through how many more inches than point A does point B travel? Express your answer as a common fraction in terms of $\pi$. <image1>
Answer: \frac{3}{4}\pi{inches}
Problem: <image>Question: A right circular cone is inscribed in a right circular cylinder. The volume of the cylinder is $72\pi$ cubic centimeters. What is the number of cubic centimeters in the space inside the cylinder but outside the cone? Express your answer in terms of $\pi$.

<image1>
Answer: 48\pi
Problem: <image>Question: In right triangle $ABC$, $M$ and $N$ are midpoints of legs $\overline{AB}$ and $\overline{BC}$, respectively. Leg $\overline{AB}$ is 6 units long, and leg $\overline{BC}$ is 8 units long. How many square units are in the area of $\triangle APC$? <image1>
Answer: 8
Problem: <image>Question: A solid right prism $ABCDEF$ has a height of $16$ and equilateral triangles bases with side length $12,$ as shown. $ABCDEF$ is sliced with a straight cut through points $M,$ $N,$ $P,$ and $Q$ on edges $DE,$ $DF,$ $CB,$ and $CA,$ respectively. If $DM=4,$ $DN=2,$ and $CQ=8,$ determine the volume of the solid $QPCDMN.$ <image1>
Answer: \frac{224\sqrt{3}}{3}
Problem: <image>Question: Triangles $BDC$ and $ACD$ are coplanar and isosceles. If we have $m\angle ABC = 70^\circ$, what is $m\angle BAC$, in degrees?

<image1>
Answer: 35
Problem: <image>Question: What is the volume of a pyramid whose base is one face of a cube of side length $2$, and whose apex is the center of the cube? Give your answer in simplest form.

<image1>
Answer: \frac{4}{3}
Problem: <image>Question: A rectangular piece of paper $ABCD$ is folded so that edge $CD$ lies along edge $AD,$ making a crease $DP.$ It is unfolded, and then folded again so that edge $AB$ lies along edge $AD,$ making a second crease $AQ.$ The two creases meet at $R,$ forming triangles $PQR$ and $ADR$. If $AB=5\mbox{ cm}$ and $AD=8\mbox{ cm},$ what is the area of quadrilateral $DRQC,$ in $\mbox{cm}^2?$

<image1>
Answer: 11.5
Problem: <image>Question: $ABCD$ is a rectangle that is four times as long as it is wide. Point $E$ is the midpoint of $\overline{BC}$. What percent of the rectangle is shaded?

<image1>
Answer: 75
Problem: <image>Question: An isosceles trapezoid is inscribed in a semicircle as shown below, such that the three shaded regions are congruent. The radius of the semicircle is one meter. How many square meters are in the area of the trapezoid? Express your answer as a decimal to the nearest tenth.

<image1>
Answer: 1.3
Problem: <image>Question: Five points $A$, $B$, $C$, $D$, and $O$ lie on a flat field.  $A$ is directly north of $O$, $B$ is directly west of $O$, $C$ is directly south of $O$, and $D$ is directly east of $O$. The  distance between $C$ and $D$ is 140 m.  A hot-air balloon is positioned in the air at $H$ directly above $O$. The balloon is held in place by four ropes $HA$, $HB$, $HC$, and $HD$.  Rope $HC$ has length 150 m and rope $HD$ has length 130 m. <image1>

To reduce the total length of rope used, rope $HC$ and rope $HD$ are to be replaced by a single rope $HP$ where $P$ is a point on the straight line between $C$ and $D$. (The balloon remains at the same position $H$ above $O$ as described above.)  Determine the greatest length of rope that can be saved.
Answer: 160
Problem: <image>Question: In the figure, point $A$ is the center of the circle, the measure of angle $RAS$ is 74 degrees, and the measure of angle $RTB$ is 28 degrees. What is the measure of minor arc $BR$, in degrees? <image1>
Answer: 81
Problem: <image>Question: In the diagram, $AD=BD=CD$ and $\angle BCA = 40^\circ.$ What is the measure of $\angle BAC?$

<image1>
Answer: 90
Problem: <image>Question: In the diagram, what is the area of $\triangle ABC$? <image1>
Answer: 54
Problem: <image>Question: Two circles are centered at the origin, as shown.  The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle.  If $QR=3$, what is the value of $k$? <image1>
Answer: 7
Problem: <image>Question: In the diagram shown here (which is not drawn to scale), suppose that $\triangle ABC \sim \triangle PAQ$ and $\triangle ABQ \sim \triangle QCP$.  If $m\angle BAC = 70^\circ$, then compute $m\angle PQC$. <image1>
Answer: 15
Problem: <image>Question: What is the ratio of the area of triangle $BDC$ to the area of triangle $ADC$?

<image1>
Answer: \frac{1}{3}
Problem: <image>Question: In triangle $ABC$, $AB = AC = 5$ and $BC = 6$.  Let $O$ be the circumcenter of triangle $ABC$.  Find the area of triangle $OBC$.

<image1>
Answer: \frac{21}{8}
Problem: <image>Question: Triangle $ABC$ and triangle $DEF$ are congruent, isosceles right triangles. The square inscribed in triangle $ABC$ has an area of 15 square centimeters. What is the area of the square inscribed in triangle $DEF$?

<image1>
Answer: \frac{40}{3}
Problem: <image>Question: In the diagram below, $\triangle ABC$ is isosceles and its area is 240.  What is the $y$-coordinate of $A?$

<image1>
Answer: 24
Problem: <image>Question: Assume that the length of Earth's equator is exactly 25,100 miles and that the Earth is a perfect sphere. The town of Lena, Wisconsin, is at $45^{\circ}$ North Latitude, exactly halfway between the equator and the North Pole. What is the number of miles in the circumference of the circle on Earth parallel to the equator and through Lena, Wisconsin? Express your answer to the nearest hundred miles.  (You may use a calculator for this problem.)

<image1>
Answer: 17700
Problem: <image>Question: In right triangle $ABC$, shown below, $\cos{B}=\frac{6}{10}$.  What is $\tan{C}$?

<image1>
Answer: \frac{3}{4}
Problem: <image>Question: Square $ABCD$ and equilateral triangle $AED$ are coplanar and share $\overline{AD}$, as shown. What is the measure, in degrees, of angle $BAE$? <image1>
Answer: 30
Problem: <image>Question: In the figure, square $WXYZ$ has a diagonal of 12 units. Point $A$ is a midpoint of segment $WX$, segment $AB$ is perpendicular to segment $AC$ and $AB = AC.$ What is the length of segment $BC$? <image1>
Answer: 18
Problem: <image>Question: In triangle $ABC$, point $D$ is on segment $BC$, the measure of angle $BAC$ is 40 degrees, and triangle $ABD$ is a reflection of triangle $ACD$ over segment $AD$. What is the measure of angle $B$?

<image1>
Answer: 70
Problem: <image>Question: A particular right square-based pyramid has a volume of 63,960 cubic meters and a height of 30 meters. What is the number of meters in the length of the lateral height ($\overline{AB}$) of the pyramid? Express your answer to the nearest whole number.

<image1>
Answer: 50
Problem: <image>Question: In triangle $ABC$, $\angle BAC = 72^\circ$.  The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively.  Find $\angle EDF$, in degrees.

<image1>
Answer: 54
Problem: <image>Question: In isosceles triangle $ABC$, angle $BAC$ and angle $BCA$ measure 35 degrees. What is the measure of angle $CDA$? <image1>
Answer: 70
Problem: <image>Question: In $\triangle ABC$, $AC=BC$, and $m\angle BAC=40^\circ$. What is the number of degrees in angle $x$? <image1>
Answer: 140
Problem: <image>Question: The two externally tangent circles each have a radius of 1 unit. Each circle is tangent to three sides of the rectangle. What is the area of the shaded region? Express your answer in terms of $\pi$.

<image1>
Answer: 8-2\pi
Problem: <image>Question: The area of $\triangle ABC$ is 6 square centimeters. $\overline{AB}\|\overline{DE}$. $BD=4BC$. What is the number of square centimeters in the area of $\triangle CDE$? <image1>
Answer: 54
Problem: <image>Question: In the diagram, $K$, $O$ and $M$ are the centers of the three semi-circles. Also, $OC = 32$ and $CB = 36$.

<image1>

What is the area of the semi-circle with center $K$?
Answer: 1250\pi
Problem: <image>Question: The volume of the cylinder shown is $45\pi$ cubic cm. What is the height in centimeters of the cylinder? <image1>
Answer: 5
Problem: <image>Question: A semi-circle of radius 8 cm, rocks back and forth along a line.  The distance between the line on which the semi-circle sits and the line above is 12 cm.  As it rocks without slipping, the semi-circle touches the line above at two points.  (When the semi-circle hits the line above, it immediately rocks back in the other direction.)  What is the distance between these two points, in millimetres, rounded off to the nearest whole number? <image1> (Note: After finding the exact value of the desired distance, you may find a calculator useful to round this value off to the nearest whole number.)
Answer: 55