conversations listlengths 2 2 | source stringclasses 1
value | score int64 5 5 |
|---|---|---|
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $VWX$ shown, $VX = \\sqrt{13}$ and $VW = 3$. What is $\\tan V$?\n\n[asy]\npair V,W,X;\nW = (0,0);\nV = (0,3);\nX = (2,0);\ndraw(V--W--X--V);\ndraw(rightanglemark(V,W,X,7));\nlabel(\"$W$\",W,SW);\nlabel(\"$V$\",V,NW... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An isosceles triangle has side lengths 8 cm, 8 cm and 10 cm. The longest side of a similar triangle is 25 cm. What is the perimeter of the larger triangle, in centimeters? [/INST] The ratio of the length of the longest sides ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $\\sin 315^\\circ$. [/INST] Let $P$ be the point on the unit circle that is $315^\\circ$ counterclockwise from $(1,0)$, and let $D$ be the foot of the altitude from $P$ to the $x$-axis, as shown below.\n\n[asy]\n\npair... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find $AX$ in the diagram.\n\n[asy]\n\nimport markers;\n\nreal t=.56;\n\npair A=(0,0);\n\npair B=(3,2);\n\npair C=(.5,1.5);\n\npair X=t*A+(1-t)*B;\n\ndraw(C--A--B--C--X);\n\nlabel(\"$A$\",A,SW);\n\nlabel(\"$B$\",B,E);\n\nlabel(... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram below, points $A$, $B$, $C$, and $P$ are situated so that $PA=2$, $PB=3$, $PC=4$, and $BC=5$. What is the maximum possible area of $\\triangle ABC$? [asy]\ndefaultpen(linewidth(0.8)); size(150);\npair B = (0,0),... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A right triangle with legs $10$ and $8$ is similar to another right triangle with corresponding legs $x$ and $5$, respectively. What is the value of $x$? Express your answer as a decimal to the nearest hundredth. [/INST] By si... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Pentagon ABCDE has a vertical line of symmetry. What is the $y$-coordinate of vertex C so that the area of the pentagon is 40 square units? [asy]\nunitsize(2mm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\ndotfactor=4;\n\npai... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Altitudes $\\overline{AD}$ and $\\overline{BE}$ of $\\triangle ABC$ intersect at $H$. If $\\angle BAC = 46^\\circ$ and $\\angle ABC = 71^\\circ$, then what is $\\angle AHB$?\n\n[asy]\nsize(150); defaultpen(linewidth(0.8));\np... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the total volume in cubic feet of three boxes if each box is a cube with edge length 4 feet? [/INST] Each box has volume $4^3=64$ cubic feet. Thus, three boxes have volume $64\\cdot3=\\boxed{192}$ cubic feet."
},
{... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The endpoints of a diameter of circle $M$ are $(-1,-4)$ and $(-7,6)$. What are the coordinates of the center of circle $M$? Express your answer as an ordered pair. [/INST] The midpoint of a diameter of a circle is its center. ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $\\sin 45^\\circ$. [/INST] Let $P$ be the point on the unit circle that is $45^\\circ$ counterclockwise from $(1,0)$, and let $D$ be the foot of the altitude from $P$ to the $x$-axis, as shown below.\n\n[asy]\n\npair A... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $\\cos 270^\\circ$. [/INST] Rotating the point $(1,0)$ about the origin by $270^\\circ$ counterclockwise gives us the point $(0,-1)$, so $\\cos 270^\\circ = \\boxed{0}$."
},
{
"from": "gpt",
"value": "Solut... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The rectangle with vertices $(-1, y), (7, y), (-1, 3)$, and $(7, 3)$ has an area of 72 square units, and $y$ is positive. What is the value of $y$? [/INST] First we write the rectangle's side lengths in terms of the coordinate... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In rectangle $ABCD$, $P$ 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 $\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The isosceles trapezoid shown has side lengths as labeled. How long is segment AC? [asy]\nunitsize(1.5mm);\ndefaultpen(linewidth(.7pt)+fontsize(10pt));\ndotfactor=3;\n\npair A=(0,0), B=(21,0), C=(15,8), D=(6,8);\npair[] dots={... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the smallest possible perimeter, in units, of a triangle whose side-length measures are consecutive integer values? [/INST] The smallest such triangle has lengths 1, 2, and 3. However, this triangle doesn't work since ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Either increasing the radius or the height of a cylinder by six inches will result in the same volume. The original height of the cylinder is two inches. What is the original radius in inches? [/INST] Let the original radius b... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two sectors of a circle of radius $12$ overlap as shown, with $P$ and $R$ as the centers of the respective circles. Determine the area of the shaded region.\n\n[asy]\ndraw((0,0)--(10.3923,-6)--(20.7846,0)--(10.3923,6)--cycle,b... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The sides of triangle $PQR$ are tangent to a circle with center $C$ as shown. Given that $\\angle PQR = 65^\\circ$ and $\\angle QRC = 30^\\circ$, find $\\angle QPR$, in degrees.\n\n[asy]\nunitsize(1.0 cm);\n\npair Q, P, R, C;\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The figure shows two concentric circles. If the length of chord AB is 80 units and chord AB is tangent to the smaller circle, what is the area of the shaded region? Express your answer in terms of $\\pi$.\n\n[asy]\ndefaultpen(... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $\\Delta ABC$ is isosceles with $AC = BC$. If $m\\angle C = 40^{\\circ}$, what is the number of degrees in $m\\angle CBD$? [asy] pair A,B,C,D,E;\nC = dir(65); B = C + dir(-65); D = (1.5,0); E = (2,0);\ndraw(B--C--A--E); dot(D)... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the number of units in the length of diagonal $DA$ of the regular hexagon shown. Express your answer in simplest radical form. [asy]\nsize(120);\ndraw((1,0)--(3,0)--(4,1.732)--(3,3.464)--(1,3.464)--(0,1.732)--cycle);\ndra... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The shaded region consists of 16 congruent squares. If $PQ = 6$ cm, what is the area of the entire shaded region?\n\n[asy]\nfor(int i = 0; i < 5; ++i)\n{\n\nfor(int j = 0; j < 2; ++j)\n\n{\n\nfilldraw((i,j)--(i+1,j)--(i+1,j+1)... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A quadrilateral in the plane has vertices $(1, 3)$, $(1, 1)$, $(2, 1)$ and $(2006, 2007)$. How many square units is the area of the quadrilateral? [/INST] The quadrilateral is shown below: [asy]\nsize(100);\ndefaultpen(linewid... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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 $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A sphere is inscribed in a cube. What is the ratio of the volume of the inscribed sphere to the volume of the cube? Express your answer as a common fraction in terms of $\\pi$. [/INST] [asy]\nsize(60);\ndraw(Circle((6,6),4.5))... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The angles of quadrilateral $ABCD$ satisfy $\\angle A = 2\\angle B =\n3\\angle C = 4\\angle D$. What is the degree measure of $\\angle A$, rounded to the nearest whole number? [/INST] Let $x$ be the degree measure of $\\angl... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has a lower base radius of 6 inches, an upper base radius of 3 inches, and a height of 4 inches, wha... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In regular pentagon $ABCDE$, diagonal $AC$ is drawn, as shown. Given that each interior angle of a regular pentagon measures 108 degrees, what is the measure of angle $CAB$?\n\n[asy]\nsize(4cm,4cm);\ndefaultpen(linewidth(1pt)+... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The lengths of the three sides of a triangle are $7$, $x+4$ and $2x+1$. The perimeter of the triangle is 36. What is the length of the longest side of the triangle? [/INST] Since the perimeter of the triangle is 36, then $7+... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $BC = 20 \\sqrt{3}$ and $\\angle C = 30^\\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$. [/INST] We have that $D$ is the midpoi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the volume, in cubic units, of a cube whose surface area is 600 square units? [/INST] The surface area of a cube equals 6 times the area of each face (since there are 6 faces). If the cube has a sidelength of $s$, then... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, the smaller circles touch the larger circle and touch each other at the center of the larger circle. The radius of the larger circle is $6.$ What is the area of the shaded region?\n\n[asy]\nsize(100);\nimport g... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The point $O$ is the center of the circle circumscribed about $\\triangle ABC$, with $\\angle BOC = 120^{\\circ}$ and $\\angle AOB =\n140^{\\circ}$, as shown. What is the degree measure of $\\angle\nABC$?\n\n[asy]\npair A,B,C;... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, $AOB$ is a sector of a circle with $\\angle AOB=60^\\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ What is the length of $XY\n?$ [asy]\ndraw((0,0)--(12,0),black+linewidth(1));\ndraw((0... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC$? [/INST] Note that $1.5^2 + 2^2 = 2.5^2,$ so $\\triangle PED$ has a right angle at $P.$ (Alternatively, you... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, $PQ$ and $RS$ are diameters of a circle with radius 4. If $PQ$ and $RS$ are perpendicular, what is the area of the shaded region?\n\n[asy]\nsize(120);\nimport graph;\nfill((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: We have a triangle $\\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$? [/INST] Let's draw a sketch first. Since $\\triangle ABC$ is isosceles, we know that $AM$ must form a right angl... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The radius of a cylinder is doubled and its height is tripled. If its original volume was 10 cubic feet, what is its volume now, in cubic feet? [/INST] Let the cylinder's original radius and height be $r$ and $h$, respectively... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the ratio of the volume of the cone to the volume of the cylinder. Express your answer as a common fraction.\n\n[asy]\nimport solids; size(150); import three; defaultpen(linewidth(0.8)); currentprojection = orthographic(5... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Suppose that we are given 40 points equally spaced around the perimeter of a square, so that four of them are located at the vertices and the remaining points divide each side into ten congruent segments. If $P$, $Q$, and $R$... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An $\\textit{annulus}$ is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$, with $b>c$. Let $\\overline{OX}$ be a radius of the larger circle, let $\\overline{XZ}$ be tange... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The square with vertices $(-a, -a), (a, -a), (-a, a), (a, a)$ is cut by the line $y = x/2$ into congruent quadrilaterals. The perimeter of one of these congruent quadrilaterals divided by $a$ equals what? Express your answer... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: When plotted in the standard rectangular coordinate system, trapezoid $ABCD$ has vertices $A(1, -2)$, $B(1, 1)$, $C(5, 7)$ and $D(5, 1)$. What is the area of trapezoid $ABCD$? [/INST] The two bases of the trapezoids are the se... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Polygon $ABCDEF$ is a regular hexagon. What is the measure in degrees of angle $ABF$? [/INST] In triangle $ABF$, the two acute angles are equal since $AB=AF$. Also, the measure of $\\angle A$ is $180^\\circ(6-2)/6=120^\\circ$.... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In right triangle $JKL$, angle $J$ measures 60 degrees and angle $K$ measures 30 degrees. When drawn, the angle bisectors of angles $J$ and $K$ intersect at a point $M$. What is the measure of obtuse angle $JMK$?\n\n[asy]\nimp... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Segment $s_1$ has endpoints at $(4,1)$ and $(-8,5)$. Segment $s_2$ is obtained by translating $s_1$ by $2$ units to the right and $3$ units up. Find the midpoint of segment $s_2$. Express your answer as $(a,b)$ with $a$ and ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circle $T$ has a circumference of $12\\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $60^{\\circ}$, what is the length, in inches, of segment $XZ$?\n\n[asy]\nsize(150);\ndraw(Circle((0,0),13),lin... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The lengths of two sides of a triangle are 33 units and 42 units. The third side also has an integral length. What is the least possible number of units in the perimeter of the triangle? [/INST] The sum of the smaller two side... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have? [asy]\n/* ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the trapezoid shown, the ratio of the area of triangle $ABC$ to the area of triangle $ADC$ is $7:3$. If $AB + CD = 210$ cm, how long is segment $\\overline{AB}$? [asy]\nimport olympiad; size(150); defaultpen(linewidth(0.8))... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, if $\\angle PQR = 48^\\circ$, what is the measure of $\\angle PMN$? [asy]\n\nsize(6cm);\n\npair p = (0, 0); pair m = dir(180 - 24); pair n = dir(180 + 24); pair r = 1.3 * dir(24); pair q = 2 * 1.3 * Cos(48) * d... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A bottled water company has designed a new cup for its dispenser. The cup will be a right circular cone with a three-inch radius. How tall will the cup need to be to hold 93 cubic inches of water? Express your answer to the ne... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The truncated right circular cone has a large base radius 8 cm and a small base radius of 4 cm. The height of the truncated cone is 6 cm. How many $\\text{cm}^3$ are in the volume of this solid? [asy]\n\nimport olympiad; size... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A right pyramid has a square base that measures 10 cm on each side. Its peak is 12 cm above the center of its base. What is the sum of the lengths of the pyramid's eight edges? Express your answer to the nearest whole number.\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The lines $y=2$, $y=5$, $x=1$, and $x=a$ make a square. Find the product of the possible values for $a$. [/INST] Since the two horizontal lines are $5-2=3$ units apart, we know that the side of the square is $3$ units long. Th... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A right circular cone is inscribed in a right prism as shown. What is the ratio of the volume of the cone to the volume of the prism? Express your answer as a common fraction in terms of $\\pi$. [asy]\nimport three; import gra... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The lengths of the sides of a triangle are 6 cm, 7 cm and 9 cm. In a similar triangle whose perimeter is 110 cm, what is the length of the longest side, in centimeters? [/INST] Let the ratio of side lengths between the simila... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the figure below, the smaller circle has a radius of two feet and the larger circle has a radius of four feet. What is the total area of the four shaded regions? Express your answer as a decimal to the nearest tenth.\n\n[as... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The measure of angle $ACB$ is 40 degrees. If ray $CA$ is rotated 480 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees?\n\n[asy]\ndraw((0,0)--dir(40),l... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, $BP$ and $BQ$ trisect $\\angle ABC$. $BM$ bisects $\\angle PBQ$. Find the ratio of the measure of $\\angle MBQ$ to the measure of $\\angle ABQ$. [/INST] Let $\\angle MBQ = x$, so $\\angle MBP=x$ as well. There... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Six congruent circles form a ring with each circle externally tangent to the two circles adjacent to it. All six circles are internally tangent to a circle $\\cal C$ with radius 30. Let $K$ be the area of the region inside $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle XYZ$, we have $\\angle X = 90^\\circ$ and $\\tan Z = 3$. What is $\\cos Z$? [/INST] [asy]\npair X,Y,Z;\nX = (0,0);\nY = (15,0);\nZ = (0,5);\ndraw(X--Y--Z--X);\ndraw(rightanglemark(Y,X,Z,23));\nlabel(\"$X$\",X,S... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Semicircles of diameter 2'' are lined up as shown. What is the area, in square inches, of the shaded region in a 1-foot length of this pattern? Express your answer in terms of $\\pi$.\n\n[asy]import graph;\nsize(101);\n\npath ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A spherical scoop of vanilla ice cream with radius of 2 inches is dropped onto the surface of a dish of hot chocolate sauce. As it melts, the ice cream spreads out uniformly forming a cylindrical region 8 inches in radius. Ass... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A delicious circular pie with diameter $12\\text{ cm}$ is cut into three equal-sized sector-shaped pieces. Let $l$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $\\cos 90^\\circ$. [/INST] Rotating the point $(1,0)$ about the origin by $90^\\circ$ counterclockwise gives us the point $(0,1)$, so $\\cos 90^\\circ = \\boxed{0}$."
},
{
"from": "gpt",
"value": "Solution ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In right triangle $MNO$, $\\tan{M}=\\frac{5}{4}$, $OM=8$, and $\\angle O = 90^\\circ$. Find $MN$. Express your answer in simplest radical form. [/INST] Our triangle is as shown below.\n\n[asy]\nsize(100);\ndraw((0,0)--(8,0)-... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 2$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A circular spinner for a game has a radius of 5 cm. The probability of winning on one spin of this spinner is $\\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\\pi$.\n\n[asy]impor... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A triangular region is enclosed by the lines with equations $y = \\frac{1}{2} x + 3$, $y = -2x + 6$ and $y = 1$. What is the area of the triangular region? Express your answer as a decimal to the nearest hundredth. [/INST] The... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $\\triangle ABC$ is similar to $\\triangle DEF$ . What is the number of centimeters in the length of $\\overline{EF}$ ? Express your answer as a decimal to the nearest tenth.\n\n[asy]\ndraw((0,0)--(8,-2)--(5,4)--cycle);\nlabel... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Three coplanar squares with sides of lengths two, four and six units, respectively, are arranged side-by-side, as shown so that one side of each square lies on line $AB$ and a segment connects the bottom left corner of the sma... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: We have a triangle $\\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude of $\\triangle ABC$. If $AC = 10,$ $BK = 7$, and $BC = 13,$ then what is the area of $\\triangle ABC$? [/INST] First, we sketch! [asy]\n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An ice cream cone has radius 1 inch and height 4 inches, What is the number of inches in the radius of a sphere of ice cream which has the same volume as the cone? [/INST] A cone with radius $r$ and height $h$ has volume $\\fr... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the two concentric circles shown, the radius of the outer circle is twice the radius of the inner circle. What is the area of the gray region, in square feet, if the width of the gray region is 2 feet? Express your answer i... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $\\sin 300^\\circ$. [/INST] Let $P$ be the point on the unit circle that is $300^\\circ$ counterclockwise from $(1,0)$, and let $D$ be the foot of the altitude from $P$ to the $x$-axis, as shown below.\n\n[asy]\npair A... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, points $U$, $V$, $W$, $X$, $Y$, and $Z$ lie on a straight line with $UV=VW=WX=XY=YZ=5$. Semicircles with diameters $UZ$, $UV$, $VW$, $WX$, $XY$, and $YZ$ create the shape shown. What is the area of the shaded... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A section is cut out of a circular piece of paper having radius four inches, as shown. Points A and B are then glued together to form a right circular cone. What is the circumference of the base of the resulting cone? Express ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the volume in cubic inches of a right, rectangular prism with side, front and bottom faces having an area 15 square inches, 10 square inches and 6 square inches, respectively? [/INST] Let the sides of the prism have le... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In right $\\Delta ABC$, $\\angle CAB$ is a right angle. Point $M$ is the midpoint of $\\overline{BC}$. What is the number of centimeters in the length of median $\\overline{AM}$? Express your answer as a decimal to the nearest... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Given that $m \\angle A= 60^\\circ$, $BC=12$ units, $\\overline{BD} \\perp \\overline{AC}$, $\\overline{CE} \\perp \\overline{AB}$ and $m \\angle DBC = 3m \\angle ECB$, the length of segment $EC$ can be expressed in the form $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Parallelogram $ABCD$ has vertices $A(3,3)$, $B(-3,-3)$, $C(-9,-3)$, and $D(-3,3)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not above the $x$-a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many square units are in the area of the parallelogram with vertices at (0, 0), (6, 0), (2, 8) and (8, 8)? [/INST] The measure of the base of the parallelogram is 6 units and the height of the parallelogram is 8 units. Th... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter of $BC$ meets side $AC$ at $D.$ If the area of $\\triangle ABC$ is $150$ and $AC = 25,$ then what is $BD$? [/INST] We might try sk... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In convex quadrilateral $ABCD$, $AB=BC=13$, $CD=DA=24$, and $\\angle D=60^\\circ$. Points $X$ and $Y$ are the midpoints of $\\overline{BC}$ and $\\overline{DA}$ respectively. Compute $XY^2$ (the square of the length of $XY$)... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The surface area of a sphere with radius $r$ is $4\\pi r^2$. Including the area of its circular base, what is the total surface area of a hemisphere with radius 6 cm? Express your answer in terms of $\\pi$.\n\n[asy]\n\nimport ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A fly is on the edge of a ceiling of a circular room with a radius of 58 feet. The fly walks straight across the ceiling to the opposite edge, passing through the center of the circle. It then walks straight to another point o... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the ratio of the volume of a cube with edge length six inches to the volume of a cube with edge length one foot? Express your answer as a common fraction. [/INST] Each side of the smaller cube is half that of the large... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Line $l_1$ has equation $3x - 2y = 1$ and goes through $A = (-1, -2)$. Line $l_2$ has equation $y = 1$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and meets $l_2$ at point $C$. The... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle XYZ$, we have $\\angle X = 90^\\circ$ and $\\tan Z = 7$. If $YZ = 100$, then what is $XY$? [/INST] [asy]\n\npair X,Y,Z;\n\nX = (0,0);\n\nY = (14,0);\n\nZ = (0,2);\n\ndraw(X--Y--Z--X);\n\ndraw(rightanglemark(Y,X... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has a perimeter of 2007 units. The sides have lengths that are all integer values with $AB< BC \\leq AC$. What is the smallest possible value of $BC - AB$? [/INST] Since $AB$ and $BC$ are positive integers and $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circle $T$ has its center at point $T(-2,6)$. Circle $T$ is reflected across the $y$-axis and then translated 8 units down. What are the coordinates of the image of the center of circle $T$? [/INST] Since the image is reflecte... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $AB = 9$, $BC = 12$, $AC = 15$, and $CD$ is the angle bisector. Find the length of $CD$. [/INST] $\\triangle ABC$ is a right triangle, since $9^2 + 12^2 = 15^2$. Thus, $\\angle ABC = 90^\\circ$.\n\n[asy]\nu... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A circle has a radius of three inches. The distance from the center of the circle to chord $CD$ is two inches. How many inches long is chord $CD$? Express your answer in simplest radical form. [/INST] Call the center of the ci... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circles $A,B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius 1 and passes through the center of $D$. What is the radius of circle $B... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Square $BCFE$ is inscribed in right triangle $AGD$, as shown below. If $AB = 28$ units and $CD = 58$ units, what is the area of square $BCFE$?\n\n[asy]\ndraw((9,15.6)--(10.7,14.6)--(11.7,16.3));\ndraw((0,0)--(10,17.3)--(40,0)-... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Points $A$, $B$, $C$, and $T$ are in space such that each of $\\overline{TA}$, $\\overline{TB}$, and $\\overline{TC}$ is perpendicular to the other two. If $TA = TB = 10$ and $TC = 9$, then what is the volume of pyramid $TABC... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A semicircle of diameter 1 sits at the top of a semicircle of diameter 2, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a $\\textit{lune}$. Determine the area of this lune.... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13,$ $19,$ $20,$ $25$ and $31,$ in some order. Find the area of the pentagon. [/INST] Let the ... | custom_source | 5 |
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