Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: In the diagram below, a triangular array of three congruent squares is configured such that the top row has one square and the bottom row has two squares. The top square lies on the two squares immediately below it. Suppose that the area of the triangle whose vertices are the centers of the three squares is $100.$ Find the area of one of the squares.\n<image1>
Answer: 200
Problem: <image>Question: Concave pentagon $ABCDE$ has a reflex angle at $D$, with $m\angle EDC = 255^o$. We are also told that $BC = DE$, $m\angle BCD = 45^o$, $CD = 13$, $AB + AE = 29$, and $m\angle BAE = 60^o$. The area of $ABCDE$ can be expressed in simplest radical form as $a\sqrt{b}$. Compute $a + b$.\n<image1>
Answer: 59
Problem: <image>Question: In the figure below, every inscribed triangle has vertices that are on the midpoints of its circumscribed triangle's sides. If the area of the largest triangle is $64$, what is the area of the shaded region?\n<image1>
Answer: 15
Problem: <image>Question: Let there be a unit square initially tiled with four congruent shaded equilateral triangles, as seen below. The total area of all of the shaded regions can be expressed in the form $\frac{a-b\sqrt{c}}{d}$ , where $a, b, c$, and $d$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c + d$.\n<image1>
Answer: 14
Problem: <image>Question: Points $ABCDEF$ are evenly spaced on a unit circle and line segments $AD$, $DF$, $FB$, $BE$, $EC$, $CA$ are drawn. The line segments intersect each other at seven points inside the circle. Denote these intersections $p_1$, $p_2$, $...$,$p_7$, where $p_7$ is the center of the circle. What is the area of the $12$-sided shape $A_{p_1}B_{p_2}C_{p_3}D_{p_4}E_{p_5}F_{p_6}$?\n<image1>
Answer: $\frac{5 \sqrt{3}}{6}$
Problem: <image>Question: Let $\vartriangle ABC$ be equilateral. Two points $D$ and $E$ are on side $BC$ (with order $B, D, E, C$), and satisfy $\angle DAE = 30^o$ . If $BD = 2$ and $CE = 3$, what is $BC$?\n<image1>
Answer: $5+\sqrt{19}$
Problem: <image>Question: Two parallel lines $\ell_1$ and $\ell_2$ lie on a plane, distance $d$ apart. On $\ell_1$ there are an infinite number of points $A_1, A_2, A_3, ...$ , in that order, with $A_nA_{n+1} = 2$ for all $n$. On $\ell_2$ there are an infinite number of points $B_1, B_2, B_3,...$ , in that order and in the same direction, satisfying $B_nB_{n+1} = 1$ for all $n$. Given that $A_1B_1$ is perpendicular to both $\ell_1$ and $\ell_2$, express the sum $\sum_{i=1}^{\infty} \angle A_iB_iA_{i+1}$ in terms of $d$.\n<image1>
Answer: $\pi-\tan ^{-1}\left(\frac{1}{d}\right)$
Problem: <image>Question: Equilateral triangle $ABC$ has $AD = DB = FG = AE = EC = 4$ and $BF = GC = 2$. From $D$ and $G$ are drawn perpendiculars to $EF$ intersecting at $H$ and $I$, respectively. The three polygons $ECGI$, $FGI$, and $BFHD$ are rearranged to $EANL$, $MNK$, and $AMJD$ so that the rectangle $HLKJ$ is formed. Find its area.\n<image1>
Answer: $16 \sqrt{3}$
Problem: <image>Question: Line $DE$ cuts through triangle $ABC$, with $DF$ parallel to $BE$. Given that $BD =DF = 10$ and $AD = BE = 25$, find $BC$.\n<image1>
Answer: 14
Problem: <image>Question: In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right?\n<image1>
Answer: 372
Problem: <image>Question: A circle inscribed in a square. Has two chords as shown in a pair. It has radius $2$, and $P$ bisects $TU$. The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle.\n<image1>
Answer: $2(\sqrt{2}-1)$
Problem: <image>Question: A Sudoku matrix is defined as a $ 9\times9$ array with entries from $ \{1, 2, \ldots , 9\}$ and with the constraint that each row, each column, and each of the nine $ 3 \times 3$ boxes that tile the array contains each digit from $ 1$ to $ 9$ exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of the squares in this matrix, as shown. What is the probability that the square marked by ? contains the digit $ 3$?\n<image1>
Answer: $\frac{2}{21}$
Problem: <image>Question: Let $ P_1,P_2,\ldots,P_8$ be $ 8$ distinct points on a circle. Determine the number of possible configurations made by drawing a set of line segments connecting pairs of these $ 8$ points, such that: $ (1)$ each $ P_i$ is the endpoint of at most one segment and $ (2)$ no two segments intersect. (The configuration with no edges drawn is allowed. An example of a valid configuration is shown below.)\n<image1>
Answer: 323
Problem: <image>Question: Let $ ABC$ be a triangle with $ \angle BAC = 90^\circ$. A circle is tangent to the sides $ AB$ and $ AC$ at $ X$ and $ Y$ respectively, such that the points on the circle diametrically opposite $ X$ and $ Y$ both lie on the side $ BC$. Given that $ AB = 6$, find the area of the portion of the circle that lies outside the triangle.\n<image1>
Answer: $\pi-2$
Problem: <image>Question: Determine the number of non-degenerate rectangles whose edges lie completely on the grid lines of the following figure.\n<image1>
Answer: 297
Problem: <image>Question: Let $ ABC$ be a triangle with $ AB = 5$, $ BC = 4$ and $ AC = 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares.\n<image1>
Answer: $\boxed{\frac{144}{49}}$
Problem: <image>Question: A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid.\n<image1>
Answer: 55/2
Problem: <image>Question: Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown. Pro selects a point $P$ at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.\n<image1>
Answer: $\boxed{\frac{3}{4}}$
Problem: <image>Question: Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges. How many subsets of these seven edges form a connected figure?\n<image1>
Answer: 81
Problem: <image>Question: Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges. Compute the number of ways to choose one or more of the seven edges such that the resulting figure is traceable without lifting a pencil. (Rotations and reflections are considered distinct.)\n<image1>
Answer: 61
Problem: <image>Question: Sam spends his days walking around the following $2\times 2$ grid of squares. Say that two squares are adjacent if they share a side. He starts at the square labeled $1$ and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal to $20$ (not counting the square he started on)?\n<image1>
Answer: 167
Problem: <image>Question: Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)\n<image1>
Answer: 18
Problem: <image>Question: Consider the L-shaped tromino below with 3 attached unit squares. It is cut into exactly two pieces of equal area by a line segment whose endpoints lie on the perimeter of the tromino. What is the longest possible length of the line segment?\n<image1>
Answer: 2.5
Problem: <image>Question: Rectangle $R_0$ has sides of lengths $3$ and $4$. Rectangles $R_1$, $R_2$, and $R_3$ are formed such that:\n$\bullet$ all four rectangles share a common vertex $P$,\n$\bullet$ for each $n = 1, 2, 3$, one side of $R_n$ is a diagonal of $R_{n-1}$,\n$\bullet$ for each $n = 1, 2, 3$, the opposite side of $R_n$ passes through a vertex of $R_{n-1}$ such that the center of $R_n$ is located counterclockwise of the center of $R_{n-1}$ with respect to $P$.\nCompute the total area covered by the union of the four rectangles.\n<image1>
Answer: 30
Problem: <image>Question: The following diagonal is drawn in a regular decagon, creating an octagon and a quadrilateral.  What is the measure of $x$?

<image1>
Answer: 36
Problem: <image>Question: In the diagram, the two triangles shown have parallel bases. What is the ratio of the area of the smaller triangle to the area of the larger triangle? <image1>
Answer: \frac{4}{25}
Problem: <image>Question: A square has a side length of 10 inches. Congruent isosceles right triangles are cut off each corner so that the resulting octagon has equal side lengths. How many inches are in the length of one side of the octagon? Express your answer as a decimal to the nearest hundredth. <image1>
Answer: 4.14
Problem: <image>Question: Three congruent isosceles triangles $DAO,$ $AOB,$ and $OBC$ have $AD=AO=OB=BC=10$ and $AB=DO=OC=12.$ These triangles are arranged to form trapezoid $ABCD,$ as shown. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB.$ <image1> What is the length of $OP?$
Answer: 8
Problem: <image>Question: In the diagram, if $\triangle ABC$ and $\triangle PQR$ are equilateral, then what is the measure of $\angle CXY$ in degrees? <image1>
Answer: 40
Problem: <image>Question: Let $ABCD$ be a parallelogram. We have that $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC.$ The segments $DM$ and $DN$ intersect $AC$ at $P$ and $Q$, respectively. If $AC = 15,$ what is $QA$? <image1>
Answer: 10
Problem: <image>Question: Corner $A$ of a rectangular piece of paper of width 8 inches is folded over so that it coincides with point $C$ on the opposite side. If $BC = 5$ inches, find the length in inches of fold $l$.

<image1>
Answer: 5\sqrt{5}
Problem: <image>Question: In the figure below, quadrilateral $CDEG$ is a square with $CD = 3$, and quadrilateral $BEFH$ is a rectangle. If $BE = 5$, how many units is $BH$? Express your answer as a mixed number. <image1>
Answer: \frac{9}{5}
Problem: <image>Question: There are two different isosceles triangles whose side lengths are integers and whose areas are $120.$ One of these two triangles, $\triangle XYZ,$ is shown. Determine the perimeter of the second triangle.

<image1>
Answer: 50
Problem: <image>Question: The measure of one of the smaller base angles of an isosceles trapezoid is $60^\circ$. The shorter base is 5 inches long and the altitude is $2 \sqrt{3}$ inches long. What is the number of inches in the perimeter of the trapezoid? <image1>
Answer: 22
Problem: <image>Question: In $\triangle{ABC}$, shown, $\cos{B}=\frac{3}{5}$.  What is $\cos{C}$?

<image1>
Answer: \frac{4}{5}
Problem: <image>Question: In the diagram, $\triangle PQR$ is isosceles.  What is the value of $x$? <image1>
Answer: 70
Problem: <image>Question: In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown. <image1>

The radius of the circle with center $R$ is decreased so that

$\bullet$ the circle with center $R$ remains tangent to $BC$,

$\bullet$ the circle with center $R$ remains tangent to the other three circles, and

$\bullet$ the circle with center $P$ becomes tangent to the other three circles.

The radii and tangencies of the other three circles stay the same. This changes the size and shape of $\triangle ABC$. $r$ is the new radius of the circle with center $R$. $r$ is of the form $\frac{a+\sqrt{b}}{c}$. Find $a+b+c$.
Answer: 6
Problem: <image>Question: In the diagram below, $WXYZ$ is a trapezoid such that $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$.  If $YZ = 12$, $\tan Z = 1.5$, and $\tan X = 3$, then what is the area of $WXYZ$?

<image1>
Answer: 162
Problem: <image>Question: In the diagram, two circles, each with center $D$, have radii of $1$ and $2$. The total area of the shaded region is $\frac{5}{12}$ of the area of the larger circle. How many degrees are in the measure of (the smaller) $\angle ADC$?
<image1>
Answer: 120
Problem: <image>Question: Three congruent isosceles triangles $DAO$, $AOB$ and $OBC$ have $AD=AO=OB=BC=10$ and $AB=DO=OC=12$.  These triangles are arranged to form trapezoid $ABCD$, as shown.  Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$.

<image1>

What is the area of trapezoid $ABCD$?
Answer: 144
Problem: <image>Question: In the diagram, $\triangle ABC$ is right-angled at $C$. Also, points $M$, $N$ and $P$ are the midpoints of sides $BC$, $AC$ and $AB$, respectively.  If the area of $\triangle APN$ is $2\mbox{ cm}^2$, then what is the area, in square centimeters, of $\triangle ABC$? <image1>
Answer: 8
Problem: <image>Question: In the diagram, what is the perimeter of the sector of the circle with radius 12?

<image1>
Answer: 24+4\pi
Problem: <image>Question: In rectangle $ABCD$ with $AB = 16,$ $P$ is a point on $BC$ so that $\angle APD=90^{\circ}$. $TS$ is perpendicular to $BC$ with $BP=PT$, as shown.  $PD$ intersects $TS$ at $Q$.  Point $R$ is on $CD$ such that $RA$ passes through $Q$.  In $\triangle PQA$, $PA=20$, $AQ=25$ and $QP=15$. <image1> Find $QR - RD$.
Answer: 0
Problem: <image>Question: A circle with center $C$ is shown. Express the area of the circle in terms of $\pi$. <image1>
Answer: 25\pi
Problem: <image>Question: In acute triangle $ABC$, altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$.  If $BD = 5$, $CD = 9$, and $CE = 42/5$, then find the length of $HE$.

<image1>
Answer: \frac{99}{20}
Problem: <image>Question: In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown. <image1>


What is the degree measure of the smallest angle in triangle $PQS$?
Answer: 30
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> Line $l$ is drawn to touch the smaller semi-circles at points $S$ and $E$ so that $KS$ and $ME$ are both perpendicular to $l$.  Determine the area of quadrilateral $KSEM$.
Answer: 2040
Problem: <image>Question: The figure below consists of four semicircles and the 16-cm diameter of the largest semicircle. What is the total number of square cm in the area of the two shaded regions? Use 3.14 as an approximation for $\pi$, and express your answer as a decimal to the nearest tenth.

<image1>
Answer: 62.8
Problem: <image>Question: A belt is drawn tightly around three circles of radius $10$ cm each, as shown. The length of the belt, in cm, can be written in the form $a + b\pi$ for rational numbers $a$ and $b$. What is the value of $a + b$? <image1>
Answer: 80
Problem: <image>Question: The point $A(3,3)$ is reflected across the $x$-axis to $A^{'}$. Then $A^{'}$ is translated two units to the left to $A^{''}$. The coordinates of $A^{''}$ are $(x,y)$. What is the value of $x+y$? <image1>
Answer: -2
Problem: <image>Question: Two right triangles share a side as follows: <image1> What is the area of $\triangle ABE$?
Answer: \frac{40}{9}
Problem: <image>Question: In the figure below, isosceles $\triangle ABC$ with base $\overline{AB}$ has altitude $CH = 24$ cm. $DE = GF$, $HF = 12$ cm, and $FB = 6$ cm. What is the number of square centimeters in the area of pentagon $CDEFG$? <image1>
Answer: 384
Problem: <image>Question: Find $AX$ in the diagram if $CX$ bisects $\angle ACB$. <image1>
Answer: 14
Problem: <image>Question: A cube of edge length $s > 0$ has the property that its surface area is equal to the sum of its volume and five times its edge length. Compute the sum of all possible values of $s$.

<image1>
Answer: 6
Problem: <image>Question: In acute triangle $ABC$, $\angle A = 68^\circ$.  Let $O$ be the circumcenter of triangle $ABC$.  Find $\angle OBC$, in degrees.

<image1>
Answer: 22
Problem: <image>Question: In the diagram below, we have $\sin \angle RPQ = \frac{7}{25}$.  What is $\cos \angle RPS$?

<image1>
Answer: -\frac{24}{25}
Problem: <image>Question: In the diagram, four squares of side length 2 are placed in the corners of a square of side length 6.  Each of the points $W$, $X$, $Y$, and $Z$ is a vertex of one of the small squares.  Square $ABCD$ can be constructed with sides passing through $W$, $X$, $Y$, and $Z$. What is the maximum possible distance from $A$ to $P$? <image1>
Answer: 6
Problem: <image>Question: The grid below contains the $16$ points whose $x$- and $y$-coordinates are in the set $\{0,1,2,3\}$: <image1> A square with all four of its vertices among these $16$ points has area $A$. What is the sum of all possible values of $A$?
Answer: 21
Problem: <image>Question: Points $A,$ $B,$ and $C$ are placed on a circle centered at $O$ as in the following diagram: <image1> If $AC = BC$ and $\angle OAC = 18^\circ,$ then how many degrees are in $\angle AOB$?
Answer: 72
Problem: <image>Question: A solid $5\times 5\times 5$ cube is composed of unit cubes. Each face of the large, solid cube is partially painted with gray paint, as shown. <image1> 	 	What fraction of the entire solid cube's unit cubes have no paint on them?
Answer: \frac{69}{125}
Problem: <image>Question: $\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?

<image1>
Answer: 28
Problem: <image>Question: Each triangle in this figure is an isosceles right triangle. The length of $\overline{BC}$ is 2 units. What is the number of units in the perimeter of quadrilateral $ABCD$? Express your answer in simplest radical form.

<image1>
Answer: 4+\sqrt{2}
Problem: <image>Question: Given regular pentagon $ABCDE,$ a circle can be drawn that is tangent to $\overline{DC}$ at $D$ and to $\overline{AB}$ at $A.$ In degrees, what is the measure of minor arc $AD$? <image1>
Answer: 144
Problem: <image>Question: In the diagram, the four points have coordinates $A(0,1)$, $B(1,3)$, $C(5,2)$, and $D(4,0)$.  What is the area of quadrilateral $ABCD$? <image1>
Answer: 9
Problem: <image>Question: Four diagonals of a regular octagon with side length 2 intersect as shown.  Find the area of the shaded region. <image1>
Answer: 4\sqrt{2}
Problem: <image>Question: In right $\triangle ABC$, shown here, $AB = 15 \text{ units}$, $AC = 24 \text{ units}$ and points $D,$ $E,$ and $F$ are the midpoints of $\overline{AC}, \overline{AB}$ and $\overline{BC}$, respectively. In square units, what is the area of $\triangle DEF$?

<image1>
Answer: 45^2
Problem: <image>Question: Each of $\triangle PQR$ and $\triangle STU$ has an area of $1.$ In $\triangle PQR,$ $U,$ $W,$ and $V$ are the midpoints of the sides. In $\triangle STU,$ $R,$ $V,$ and $W$ are the midpoints of the sides. What is the area of parallelogram $UVRW?$ <image1>
Answer: \frac{1}{2}
Problem: <image>Question: If the area of the triangle shown is 40, what is $r$? <image1>
Answer: 10
Problem: <image>Question: An 8-inch by 8-inch square is folded along a diagonal creating a triangular region. This resulting triangular region is then folded so that the right angle vertex just meets the midpoint of the hypotenuse. What is the area of the resulting trapezoidal figure in square inches?

<image1>
Answer: 24
Problem: <image>Question: Elliott Farms has a silo for storage. The silo is a right circular cylinder topped by a right circular cone, both having the same radius. The height of the cone is half the height of the cylinder. The diameter of the base of the silo is 10 meters and the height of the entire silo is 27 meters. What is the volume, in cubic meters, of the silo? Express your answer in terms of $\pi$.

<image1>
Answer: 525\pi
Problem: <image>Question: In $\triangle ABC$, what is the value of $x + y$? <image1>
Answer: 90
Problem: <image>Question: In rectangle $ABCD$, $AD=1$, $P$ is on $\overline{AB}$, and $\overline{DB}$ and $\overline{DP}$ trisect $\angle ADC$. Write the perimeter of $\triangle BDP$ in simplest form as:  $w + \frac{x \cdot \sqrt{y}}{z}$, where $w, x, y, z$ are nonnegative integers.  What is $w + x + y + z$?

<image1>
Answer: 12
Problem: <image>Question: In the figure below $AB = BC$, $m \angle ABD = 30^{\circ}$, $m \angle C = 50^{\circ}$ and $m \angle CBD = 80^{\circ}$. What is the number of degrees in the measure of angle $A$?

<image1>
Answer: 75
Problem: <image>Question: In regular pentagon $PQRST$, $X$ is the midpoint of segment $ST$. What is the measure of angle $XQS,$ in degrees?

<image1>
Answer: 18
Problem: <image>Question: In isosceles triangle $ABC$, if $BC$ is extended to a point $X$ such that $AC = CX$, what is the number of degrees in the measure of angle $AXC$? <image1>
Answer: 15
Problem: <image>Question: A right hexagonal prism has a height of 3 feet and each edge of the hexagonal bases is 6 inches. What is the sum of the areas of the non-hexagonal faces of the prism, in square feet?

<image1>
Answer: 9
Problem: <image>Question: $ABCD$ is a square 4 inches on a side, and each of the inside squares is formed by joining the midpoints of the outer square's sides. What is the area of the shaded region in square inches?

<image1>
Answer: 4
Problem: <image>Question: A hexagon is drawn with its vertices at $$(0,0),(1,0),(2,1),(2,2),(1,2), \text{ and } (0,1),$$ and all of its diagonals are also drawn, as shown below. The diagonals cut the hexagon into $24$ regions of various shapes and sizes. These $24$ regions are shown in pink and yellow below. If the smallest region (by area) has area $a$, and the largest has area $b$, then what is the ratio $a:b$? Give your answer in lowest terms. <image1>
Answer: 1:2
Problem: <image>Question: In the circle with center $O$ and diameters $AC$ and $BD$, the angle $AOD$ measures $54$ degrees. What is the measure, in degrees, of angle $AOB$? <image1>
Answer: 126
Problem: <image>Question: The perimeter of $\triangle ABC$ is $32.$ If $\angle ABC=\angle ACB$ and $BC=12,$ what is the length of $AB?$

<image1>
Answer: 10
Problem: <image>Question: In circle $J$, $HO$ and $HN$ are tangent to the circle at $O$ and $N$. Find the number of degrees in the sum of $m\angle J$ and $m\angle H$. <image1>
Answer: 180
Problem: <image>Question: A sphere is inscribed in a cone with height 4 and base radius 3.  What is the ratio of the volume of the sphere to the volume of the cone?

<image1>
Answer: \frac{3}{8}
Problem: <image>Question: The length of the diameter of this spherical ball is equal to the height of the box in which it is placed. The box is a cube and has an edge length of 30 cm. How many cubic centimeters of the box are not occupied by the solid sphere? Express your answer in terms of $\pi$. <image1>
Answer: 27000-4500\pi
Problem: <image>Question: In the circle below, $\overline{AB} \| \overline{CD}$. $\overline{AD}$ is a diameter of the circle, and $AD = 36^{\prime \prime}$. What is the number of inches in the length of $\widehat{AB}$? Express your answer in terms of $\pi$. <image1>
Answer: 8\pi
Problem: <image>Question: In right triangle $ABC$, $\angle B = 90^\circ$, and $D$ and $E$ lie on $AC$ such that $\overline{BD}$ is a median and $\overline{BE}$ is an altitude.  If $BD=2\cdot DE$, compute $\frac{AB}{EC}$. <image1>
Answer: 2\sqrt{3}
Problem: <image>Question: The area of square $ABCD$ is 100 square centimeters, and $AE = 2$ cm. What is the area of square $EFGH$, in square centimeters?

<image1>
Answer: 68
Problem: <image>Question: In the diagram, $R$ is on $QS$ and $QR=8$. Also, $PR=12$, $\angle PRQ=120^\circ$, and $\angle RPS = 90^\circ$. What is the area of $\triangle QPS$? <image1>
Answer: 96\sqrt{3}