from stringclasses 2
values | value stringlengths 16 6.77k |
|---|---|
human | Triangle $ABC$ is an obtuse, isosceles triangle. Angle $A$ measures 20 degrees. What is number of degrees in the measure of the largest interior angle of triangle $ABC$?
[asy]
draw((-20,0)--(0,8)--(20,0)--cycle);
label("$20^{\circ}$",(-13,-0.7),NE);
label("$A$",(-20,0),W);
label("$B$",(0,8),N);
label("$C$",(20,0),E);
... |
gpt | Since $\triangle ABC$ is isosceles, $\angle C = 20$ degrees. Thus, $\angle B = 180 - 20 - 20 = 140$ degrees. So the largest interior angle is $\boxed{140}$ degrees. |
human | In triangle $ABC$, we have that $E$ and $F$ are midpoints of sides $\overline{AC}$ and $\overline{AB}$, respectively. The area of $\triangle ABC$ is 24 square units. How many square units are in the area of $\triangle CEF$? |
gpt | We first draw a diagram: [asy]
pair A, B, C, E, F;
A = (0, 4);
B = (-3, 0);
C = (7, 0);
E = 0.5 * A + 0.5 * C;
F = 0.5 * A + 0.5 * B;
draw(A--B--C--cycle);
draw(C--E--F--cycle);
label("$A$", A, N);
label("$B$", B, NW);
label("$C$", C, NE);
label("$E$", E, NE);
label("$F$", F, NW);
[/asy] Since $F$ is the midpoint of $\... |
human | Carla rotated point $A$ 420 degrees clockwise about point $B$ and it landed at point $C$. Devon rotated the original point $A$ $x$ degrees counterclockwise about point $B$ and it also landed at point $C$. If $x<360$, what is the value of $x$? |
gpt | Carla's rotation was equivalent to one of $60^{\circ}$ clockwise. To get to the same point by going counter-clockwise, we have to go $360^{\circ}$ minus Carla's rotation, or $\boxed{300^{\circ}}$. |
human | In the diagram, $ABCD$ and $EFGD$ are squares each of area 16. If $H$ is the midpoint of both $BC$ and $EF$, find the total area of polygon $ABHFGD$.
[asy]
unitsize(3 cm);
pair A, B, C, D, E, F, G, H;
F = (0,0);
G = (1,0);
D = (1,1);
E = (0,1);
H = (E + F)/2;
A = reflect(D,H)*(G);
B = reflect(D,H)*(F);
C = reflect(... |
gpt | Draw $DH$.
[asy]
unitsize(3 cm);
pair A, B, C, D, E, F, G, H;
F = (0,0);
G = (1,0);
D = (1,1);
E = (0,1);
H = (E + F)/2;
A = reflect(D,H)*(G);
B = reflect(D,H)*(F);
C = reflect(D,H)*(E);
draw(A--B--C--D--cycle);
draw(D--E--F--G--cycle);
draw(D--H,dashed);
label("$A$", A, N);
label("$B$", B, W);
label("$C$", C, S);... |
human | A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$? |
gpt | The side length of the triangle and hexagon are $\frac{a}{3}$ and $\frac{b}{6},$ respectively, so their areas are \[\frac{\sqrt{3}}{4} \left(\frac{a}{3}\right)^2 = \frac{a^2 \sqrt3}{36} \quad \text{and} \quad \frac{3\sqrt3}{2} \left(\frac{b}{6}\right)^2 = \frac{b^2\sqrt3}{24},\]respectively. Therefore, we have \[\frac{... |
human | Triangle $PAB$ is formed by three tangents to circle $O$ and $\angle APB = 40^\circ$. Find $\angle AOB$.
[asy]
import graph;
unitsize(1.5 cm);
pair A, B, O, P, R, S, T;
R = dir(115);
S = dir(230);
T = dir(270);
P = extension(R, R + rotate(90)*(R), T, T + rotate(90)*(T));
A = extension(S, S + rotate(90)*(S), T, T +... |
gpt | First, from triangle $ABO$, $\angle AOB = 180^\circ - \angle BAO - \angle ABO$. Note that $AO$ bisects $\angle BAT$ (to see this, draw radii from $O$ to $AB$ and $AT,$ creating two congruent right triangles), so $\angle BAO = \angle BAT/2$. Similarly, $\angle ABO = \angle ABR/2$.
Also, $\angle BAT = 180^\circ - \ang... |
human | In triangle $XYZ$, we have $\angle Z = 90^\circ$, $XY = 10$, and $YZ = \sqrt{51}$. What is $\tan X$? |
gpt | [asy]
pair X,Y,Z;
Z = (0,0);
Y = (sqrt(51),0);
X = (0,7);
draw(X--Y--Z--X);
draw(rightanglemark(Y,Z,X,15));
label("$X$",X,NE);
label("$Y$",Y,SE);
label("$Z$",Z,SW);
label("$10$",(X+Y)/2,NE);
label("$\sqrt{51}$",(Z+Y)/2,S);
[/asy]
Because this is a right triangle, $\tan X = \frac{YZ}{XZ}$.
Using the Pythag... |
human | The points $B(1, 1)$, $I(2, 4)$ and $G(5, 1)$ are plotted in the standard rectangular coordinate system to form triangle $BIG$. Triangle $BIG$ is translated five units to the left and two units upward to triangle $B'I'G'$, in such a way that $B'$ is the image of $B$, $I'$ is the image of $I$, and $G'$ is the image of $... |
gpt | Since triangle $B^\prime I^\prime G^\prime$ is translated from triangle $BIG,$ the midpoint of $B^\prime G ^\prime $ is the midpoint of $BG$ translated five units left and two units up. The midpoint of $BG$ is at $\left( \frac{1+5}{2}, \frac{1+1}{2} \right) = (3, 1).$ Thus, the midpoint of $B ^\prime G ^\prime$ is at $... |
human | How many cubic feet are in the volume of a round swimming pool which is 16 feet in diameter and 4 feet deep throughout? Express your answer in terms of $\pi$. |
gpt | The radius of this pool is $16/2=8$ feet; the volume of this pool is thus $\pi(8^2)(4)=\boxed{256\pi}$ cubic feet. |
human | Side $AB$ of regular hexagon $ABCDEF$ is extended past $B$ to point $X$ such that $AX = 3AB$. Given that each side of the hexagon is $2$ units long, what is the length of segment $FX$? Express your answer in simplest radical form. |
gpt | Let $P$ be the foot of the perpendicular from $F$ to the line containing $AB$. [asy]size(150);
defaultpen(linewidth(0.7) + fontsize(10)); real lsf = 0.6;
pair C = (2,0), B = 2*dir(60), A = 2*dir(120), F = -C, E = -B, D = -A, P = foot(F,A,B), Y = B+(4,0);
draw(A--B--C--D--E--F--cycle); draw(F--P--Y--cycle); draw(rightan... |
human | The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.
[asy]
unitsize(0.8cm);
for (int i=0; i<7; ++i) {
for (int j=0; j<4; ++j) {
dot((i,j));
};}
label("$A$",(0,3),W);
label("$B$",(6,0),E);
label("$D$",(2,0),S);
label("$E$... |
gpt | Extend $\overline{DC}$ to $F$. Triangle $FAE$ and $DBE$ are similar with ratio $5:4$. Thus $AE=\frac{5AB}{9}$, $AB=\sqrt{3^2+6^2}=\sqrt{45}=3\sqrt{5}$, and $AE=\frac{5(3\sqrt{5})}{9}=\boxed{\frac{5\sqrt{5}}{3}}$. [asy]
unitsize(0.8cm);
for (int i=0; i<7; ++i) {
for (int j=0; j<4; ++j) {
dot((i,j));
};}
label("$F$",(5,3... |
human | Two cylindrical cans have the same volume. The height of one can is triple the height of the other. If the radius of the narrower can is 12 units, how many units are in the length of the radius of the wider can? Express your answer in simplest radical form. |
gpt | Let the height of the wide can be $h$ and the height of the narrow can be $3h$. Let the wide can have radius $x$ units. Since the two volumes are equal, we have \[\pi (12^2) (3h) = \pi (x^2) (h).\] Solving yields $x = 12\sqrt{3}$ so the wide can has radius $\boxed{12\sqrt{3}}$ units. |
human | Find the number of square units in the area of the triangle.
[asy]size(125);
draw( (-10,-2) -- (2,10), Arrows);
draw( (0,-2)-- (0,10) ,Arrows);
draw( (5,0) -- (-10,0),Arrows);
label("$l$",(2,10), NE);
label("$x$", (5,0) , E);
label("$y$", (0,-2) , S);
filldraw( (-8,0) -- (0,8) -- (0,0) -- cycle, lightgray);
dot( (-2,... |
gpt | We first notice that the vertical and horizontal distances between the two points are both $4$, so the slope of the line which the two points are on must be $1$. We now find the length of the legs of the triangle. Since the slope of the line is one, we can add $2$ to both the $x$ and $y$-coordinates of $(-2,6)$ and get... |
human | Triangle $ABC$ is isosceles with angle $A$ congruent to angle $B$. The measure of angle $C$ is 30 degrees more than the measure of angle $A$. What is the number of degrees in the measure of angle $C$? |
gpt | Let $x$ be the number of degrees in the measure of angle $A$. Then angle $B$ measures $x$ degrees as well and angle $C$ measures $x+30$ degrees. Since the sum of the interior angles in a triangle sum to 180 degrees, we solve $x+x+x+30=180$ to find $x=50$. Therefore, angle $C$ measures $x+30=50+30=\boxed{80}$ degrees... |
human | Find $\tan Y$ in the right triangle shown below.
[asy]
pair X,Y,Z;
X = (0,0);
Y = (24,0);
Z = (0,7);
draw(X--Y--Z--X);
draw(rightanglemark(Y,X,Z,23));
label("$X$",X,SW);
label("$Y$",Y,SE);
label("$Z$",Z,N);
label("$25$",(Y+Z)/2,NE);
label("$24$",Y/2,S);
[/asy] |
gpt | The Pythagorean Theorem gives us $XZ= \sqrt{YZ^2 - XY^2} = \sqrt{625-576} = \sqrt{49}=7$, so $\tan Y = \frac{XZ}{XY} = \ \boxed{\frac{7}{24}}$. |
human | Find $\tan G$ in the right triangle shown below.
[asy]
pair H,F,G;
H = (0,0);
G = (15,0);
F = (0,8);
draw(F--G--H--F);
draw(rightanglemark(F,H,G,20));
label("$H$",H,SW);
label("$G$",G,SE);
label("$F$",F,N);
label("$17$",(F+G)/2,NE);
label("$15$",G/2,S);
[/asy] |
gpt | The Pythagorean Theorem gives us $FH= \sqrt{FG^2 - GH^2} = \sqrt{289-225} = \sqrt{64}=8$, so $\tan G = \frac{FH}{HG} = \ \boxed{\frac{8}{15}}$. |
human | A right square pyramid with base edges of length $8\sqrt{2}$ units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 3 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane? [asy]
import three;
size(2.5inch);
currentproje... |
gpt | Define the points $A$, $B$, $C$ , and $D$, $E$, and $F$ as shown so that $AC$ is perpendicular to the base of the pyramid. Segment $DC$ is a leg of the isosceles right triangle $CDF$ whose hypotenuse is $8\sqrt{2}$. Therefore, $CD=8\sqrt{2}/\sqrt{2}=8$. Applying the Pythagorean theorem to triangle $ACD$ gives $AC=6$... |
human | What is the area, in square units, of a regular hexagon inscribed in a circle whose area is $324\pi$ square units? Express your answer in simplest radical form. |
gpt | Note that since the area is $\pi r^2 = 324 \pi$, where $r$ is the radius, we must have $r=\sqrt{324}=18$. Thus the distance from the center of the hexagon to a vertex is $18$, and we can break up the hexagon into $6$ equilateral triangles, each of which has side length $18$. The area of an equilateral triangle of side ... |
human | Points $P$ and $R$ are located at (2, 1) and (12, 15) respectively. Point $M$ is the midpoint of segment $\overline{PR}$. Segment $\overline{PR}$ is reflected over the $x$-axis. What is the sum of the coordinates of the image of point $M$ (the midpoint of the reflected segment)? |
gpt | Point $M$ has coordinates $(7,8)$. Therefore, its image has coordinates $(7,-8)$. Thus the sum is $7-8 = \boxed{-1}$.
Alternatively, the image of point $M$ is the midpoint of the images of points $P$ and $R$ and thus is the midpoint of $(2,-1)$ and $(12,-15)$, which is also $(7,-8)$. |
human | What is the number of centimeters in the length of $EF$ if $AB\parallel CD\parallel EF$?
[asy]
size(4cm,4cm);
pair A,B,C,D,E,F,X;
A=(0,1);
B=(1,1);
C=(1,0);
X=(0,0);
D=(1/3)*C+(2/3)*X;
draw (A--B--C--D);
draw(D--B);
draw(A--C);
E=(0.6,0.4);
F=(1,0.4);
draw(E--F);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$"... |
gpt | Since $AB\parallel EF,$ we know that $\angle BAC = \angle FEC$ and $\angle ABC = \angle EFC.$ Therefore, we see that $\triangle ABC \sim \triangle EFC$ by AA Similarity. Likewise, $\triangle BDC \sim \triangle BEF.$
From our similarities, we can come up with two equations: $\dfrac{BF}{BC} = \dfrac{EF}{DC}$ and $\dfrac... |
human | A dump truck delivered sand to a construction site. The sand formed a conical pile with a diameter of $8$ feet and a height that was $75\%$ of the diameter. How many cubic feet of sand were in the pile? Express your answer in terms of $\pi$. |
gpt | The height of the cone is $\frac{3}{4} \times 8 = 6$. The radius of the cone is $\frac{8}{2} = 4$. Therefore, the volume of the cone formed is $\frac{4^2 \times 6 \times \pi}{3} = \boxed{32 \pi}$. |
human | A two-gallon container had all of its dimensions tripled. How many gallons does the new container hold? |
gpt | Suppose that our two-gallon container is in the shape of a rectangular prism. If we triple the length, the volume triples. Tripling the width or the height gives us the same result. Therefore, tripling all of the dimensions increases the volume by a factor of $3\cdot 3 \cdot 3 = 27$. The new container can hold $2 \time... |
human | The image of the point with coordinates $(-3,-1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(5,3)$. Find $m+b$. |
gpt | The line of reflection is the perpendicular bisector of the segment connecting the point with its image under the reflection. The slope of the segment is $\frac{3-(-1)}{5-(-3)}=\frac{1}{2}$. Since the line of reflection is perpendicular, its slope, $m$, equals $-2$. By the midpoint formula, the coordinates of the mi... |
human | A square pyramid has a base edge of 32 inches and an altitude of 1 foot. A square pyramid whose altitude is one-fourth of the original altitude is cut away at the apex of the original pyramid. The volume of the remaining frustum is what fractional part of the volume of the original pyramid? |
gpt | The piece that is removed from the original pyramid to create the frustum is itself a square pyramid that is similar to the original pyramid. The ratio of corresponding side lengths is 1/4, so the piece that was removed has volume $(1/4)^3 = 1/64$ of the volume of the original pyramid. Therefore, the remaining frustu... |
human | What is the perimeter of pentagon $ABCDE$ in this diagram? [asy]
pair cis(real r,real t) { return (r*cos(t),r*sin(t)); }
pair a=(0,0);
pair b=cis(1,-pi/2);
pair c=cis(sqrt(2),-pi/4);
pair d=cis(sqrt(3),-pi/4+atan(1/sqrt(2)));
pair e=cis(2,-pi/4+atan(1/sqrt(2))+atan(1/sqrt(3)));
dot(a); dot(b); dot(c); dot(d); dot(e);
d... |
gpt | By the Pythagorean theorem, we have: \begin{align*}
AC^2 &= AB^2 + BC^2 = 1+1 = 2; \\
AD^2 &= AC^2 + CD^2 = 2+1 = 3; \\
AE^2 &= AD^2 + DE^2 = 3+1 = 4.
\end{align*}Thus $AE=\sqrt 4=2,$ and the perimeter of pentagon $ABCDE$ is $1+1+1+1+2 = \boxed{6}$. |
human | Compute $\cos 0^\circ$. |
gpt | Rotating the point $(1,0)$ about the origin by $0^\circ$ counterclockwise gives us the point $(1,0)$, so $\cos 0^\circ = \boxed{1}$. |
human | The same amount of steel used to create eight solid steel balls, each with a radius of 1 inch, is used to create one larger steel ball. What is the radius of the larger ball?
[asy]
size(150);
filldraw(circle((0,0),1),gray);
filldraw(circle((.9,-.8),1),gray);
filldraw(circle((1.8,.9),1),gray);
filldraw(circle((2,0),1),... |
gpt | The amount of steel used to create one ball with radius 1 is $\frac{4}{3}\pi(1^3)=\frac{4}{3}\pi$; the amount of steel used to create eight of these balls is $8\cdot \frac{4}{3}\pi = \frac{32}{3}\pi$.
Let the radius of the large steel be $r$. We have $\frac{4}{3}\pi r^3 = \frac{32}{3}\pi$; solving for $r$ yields $r^3... |
human | What is the sum of the number of faces, edges and vertices of a triangular prism?
[asy]
draw((0,0)--(10,0)--(5,8.7)--cycle);
draw((0,0)--(20,20),dashed);
draw((10,0)--(30,20));
draw((5,8.7)--(25,28.7));
draw((25,28.7)--(30,20)--(20,20)--cycle,dashed);
draw((25,28.7)--(30,20));
[/asy] |
gpt | Faces: There are $3$ on the sides, a top face, and a bottom face, so $5$.
Edges: There are $3$ on the top, $3$ on the bottom, and $3$ connecting them, for $9$.
Vertices: There are $3$ on the top and $3$ on the bottom, for $6$.
So $5+9+6=\boxed{20}$. |
human | In right triangle $ABC$, $AB=9$, $BC=13$, and $\angle B = 90^\circ$. Points $D$ and $E$ are midpoints of $\overline{AB}$ and $\overline{AC}$ respectively; $\overline{CD}$ and $\overline{BE}$ intersect at point $X$. Compute the ratio of the area of quadrilateral $AEXD$ to the area of triangle $BXC$. |
gpt | We begin by drawing a diagram: [asy]
pair A,B,C,D,E,X;
A=(0,9); B=(0,0); C=(13,0); E=(A+C)/2; D=(A+B)/2; X = intersectionpoint(B--E,D--C); label("$X$",X,N);
fill(A--E--X--D--cycle,rgb(135,206,250));
fill(B--X--C--cycle,rgb(107,142,35));
draw(A--B--C--cycle);
draw(C--D); draw(B--E);
draw(rightanglemark(A,B,C,15));
labe... |
human | A hexagon is obtained by joining, in order, the points $(0,1)$, $(1,2)$, $(2,2)$, $(2,1)$, $(3,1)$, $(2,0)$, and $(0,1)$. The perimeter of the hexagon can be written in the form $a+b\sqrt{2}+c\sqrt{5}$, where $a$, $b$ and $c$ are whole numbers. Find $a+b+c$. |
gpt | We must find the length of each side of the hexagon to find the perimeter.
We can see that the distance between each pair of points $(1, 2)$ and $(2, 2)$, $(2, 2)$ and $(2, 1)$, and $(2, 1)$ and $(3, 1)$ is 1. Thus, these three sides have a total length of 3.
We can see that the distance between $(0, 1)$ and $(1, 2)$... |
human | The triangle $\triangle ABC$ is an isosceles triangle where $AB = 4\sqrt{2}$ and $\angle B$ is a right angle. If $I$ is the incenter of $\triangle ABC,$ then what is $BI$?
Express your answer in the form $a + b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers, and $c$ is not divisible by any perfect square other than $1... |
gpt | We might try sketching a diagram: [asy]
pair pA, pB, pC, pI;
pA = (-1, 0);
pB = (0, 0);
pC = (0, 1);
pI = (-0.2929, 0.2929);
draw(pA--pB--pC--pA);
draw(pI--pB);
draw(circle(pI, 0.2929));
label("$A$", pA, SW);
label("$B$", pB, SE);
label("$C$", pC, NE);
label("$I$", pI, NE);
[/asy] Since $\triangle ABC$ is isosceles, we... |
human | In the staircase-shaped region below, all angles that look like right angles are right angles, and each of the eight congruent sides marked with a tick mark have length 1 foot. If the region has area 53 square feet, what is the number of feet in the perimeter of the region? [asy]
size(120);
draw((5,7)--(0,7)--(0,0)--(... |
gpt | We can look at the region as a rectangle with a smaller staircase-shaped region removed from its upper-right corner. We extend two of its sides to complete the rectangle: [asy]
size(120);
draw((5,7)--(0,7)--(0,0)--(9,0)--(9,3)--(8,3)--(8,4)--(7,4)--(7,5)--(6,5)--(6,6)--(5,6)--cycle);
draw((5,7)--(9,7)--(9,3),dashed);
... |
human | One leg of a right triangle is 12 inches, and the measure of the angle opposite that leg is $30^\circ$. What is the number of inches in the hypotenuse of the triangle? |
gpt | A right triangle with a $30^\circ$ angle is a 30-60-90 triangle. In such a triangle, the hypotenuse has twice the length of the leg opposite the $30^\circ$ angle, so the hypotenuse of the triangle in the problem has length $2\cdot 12 = \boxed{24}$ inches. |
human | Let $ABCDEF$ be a regular hexagon, and let $G,H,I$ be the midpoints of sides $AB,CD,EF$ respectively. If the area of $\triangle GHI$ is $225$, what is the area of hexagon $ABCDEF$? |
gpt | We begin with a diagram of the given information: [asy]
size(4cm);
real x=sqrt(3);
pair d=(2,0); pair c=(1,x); pair b=(-1,x); pair a=-d; pair f=-c; pair e=-b;
pair g=(a+b)/2; pair h=(c+d)/2; pair i=(e+f)/2;
draw(a--b--c--d--e--f--a);
dot(a); dot(b); dot(c); dot(d); dot(e); dot(f); dot(g); dot(h); dot(i);
draw(g--h--i--... |
human | Six small circles, each of radius $3$ units, are tangent to a large circle as shown. Each small circle also is tangent to its two neighboring small circles. What is the diameter of the large circle in units? [asy]
draw(Circle((-2,0),1));
draw(Circle((2,0),1));
draw(Circle((-1,1.73205081),1));
draw(Circle((1,1.73205081)... |
gpt | We can draw two similar hexagons, an outer one for which the large circle is the circumcircle and an inner one that connects the centers of the smaller circles. We know that the sidelength of the inner hexagon is 6 since $\overline{DE}$ consists of the radii of two small circles. We also know that the radius of the out... |
human | A pyramid with volume 40 cubic inches has a rectangular base. If the length of the base is doubled, the width tripled and the height increased by $50\%$, what is the volume of the new pyramid, in cubic inches? |
gpt | Since the volume of a pyramid is linear in each of length, width, and height (in particular, $V = \frac{1}{3} lwh$), multiplying any of these dimensions by a scalar multiplies the volume by the same scalar. So the new volume is $2\cdot 3\cdot 1.50 = 9$ times the old one, or $\boxed{360}$ cubic inches. |
human | The line $y=-\frac{3}{4}x+9$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is three times the area of $\triangle TOP$, then what is the value of $r+s$? [asy]
// draw axes
draw((-3, 0)--(15, 0), Arrow); draw((0, -3)--(0, 15), Arrow);
label(... |
gpt | The $y$-intercept of the line $y = -\frac{3}{4}x+9$ is $y=9$, so $Q$ has coordinates $(0, 9)$.
To determine the $x$-intercept, we set $y=0$, and so obtain $0 = -\frac{3}{4}x+9$ or $\frac{3}{4}x=9$ or $x=12$. Thus, $P$ has coordinates $(12, 0)$.
Therefore, the area of $\triangle POQ$ is $\frac{1}{2}(12)(9) = 54$, sin... |
human | In triangle $ABC$, $AB = 7$, $AC = 15$, and the length of median $AM$ is 10. Find the area of triangle $ABC$. |
gpt | Extend $AM$ to $D$ so that $MD = MA$. Then triangles $AMB$ and $DMC$ are congruent, so triangles $ABC$ and $ACD$ have equal area.
[asy]
unitsize(0.3 cm);
pair A, B, C, D, M;
A = (-7/sqrt(37),42/sqrt(37));
B = (0,0);
C = (2*sqrt(37),0);
M = (B + C)/2;
D = 2*M - A;
draw(A--B--C--cycle);
draw(A--D--C);
label("$A$", ... |
human | The midpoints of the sides of a triangle with area $T$ are joined to form a triangle with area $M$. What is the ratio of $M$ to $T$? Express your answer as a common fraction. |
gpt | When you connect the midpoints of two sides of a triangle, you get a segment which is half as long as the third side of the triangle. Therefore, every side in the smaller triangle is $\frac{1}{2}$ the side length of the original triangle. Therefore, the area of the smaller triangle is $\left(\frac{1}{2}\right)^2 = \box... |
human | In the diagram, the area of triangle $ABC$ is 27 square units. What is the area of triangle $BCD$?
[asy]
draw((0,0)--(32,0)--(9,15)--(0,0));
dot((0,0));
label("$A$",(0,0),SW);
label("6",(3,0),S);
dot((6,0));
label("$C$",(6,0),S);
label("26",(19,0),S);
dot((32,0));
label("$D$",(32,0),SE);
dot((9,15));
label("$B$",(9,... |
gpt | Let $h$ be the distance from $B$ to side $AD$. The area of $ABC$ is 27, so $\frac{1}{2}\cdot6\cdot h = 27$, which implies $h=9$. The area of $BCD$ is $\frac{1}{2}\cdot26\cdot9=\boxed{117}$ square units. |
human | Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same nonzero number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a common fraction. |
gpt | Let the increase measure $x$ inches. The cylinder with increased radius now has volume \[\pi (8+x)^2 (3)\] and the cylinder with increased height now has volume \[\pi (8^2) (3+x).\] Setting these two quantities equal and solving yields \[3(64+16x+x^2)=64(3+x) \Rightarrow 3x^2-16x=x(3x-16)=0\] so $x=0$ or $x=16/3$. ... |
human | Let $C_1$ and $C_2$ be circles defined by $$
(x-10)^2+y^2=36
$$and $$
(x+15)^2+y^2=81,
$$respectively. What is the length of the shortest line segment $\overline{PQ}$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$? |
gpt | The centers are at $A=(10,0)$ and $B=(-15,0)$, and the radii are 6 and 9, respectively. Since the internal tangent is shorter than the external tangent, $\overline{PQ}$ intersects $\overline{AB}$ at a point $D$ that divides $\overline{AB}$ into parts proportional to the radii. The right triangles $\triangle APD$ and ... |
human | Wei has designed a logo for his new company using circles and a large square, as shown. Each circle is tangent to two sides of the square and its two adjacent circles. If he wishes to create a version of this logo that is 20 inches on each side, how many square inches will be shaded?
[asy]
size(100);
draw((0,0)--(4,... |
gpt | The shaded area is equal to the area of the square minus the area of the four circles. Since the side length of the square is $20$ inches, the radius of the circles is $20/4 = 5$ inches. The square has area $20^2 = 400$ square inches, and each circle has area $5^2 \pi = 25\pi$ square inches, so the shaded area, in sq... |
human | 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 $TA = TB = 12$ and $TC = 6$, then what is the distance from $T$ to face $ABC$? |
gpt | [asy]
import three;
triple A = (4,8,0);
triple B= (4,0,0);
triple C = (0,0,0);
triple D = (0,8,0);
triple P = (4,8,6);
draw(B--P--D--A--B);
draw(A--P);
draw(B--D,dashed);
label("$T$",A,S);
label("$B$",B,W);
label("$C$",D,E);
label("$A$",P,N);
label("$M$",(P+B)/2,NW);
draw(D--((P+B)/2),dashed);
[/asy]
We can think of $... |
human | Two circles are drawn in a 12-inch by 14-inch rectangle. Each circle has a diameter of 6 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles? |
gpt | Suppose we put the two circles in opposite corners of the rectangle so that the circles are tangent to the sides of the rectangle, and they are diagonally across from each other. Then the center of each circle is 3 inches in from each side of the rectangle that it touches. Now imagine a rectangle that has opposite corn... |
human | Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$ in the diagram if $CX$ bisects $\angle ACB$. Express your answer as a common fraction. [asy]
import markers;
real t=27/(27+30);
pair A=(-15.57,0);
pair B=(8.43,0);
pair C=(0,25.65);
pair X=t*A+(1-t)*B;
draw(C--A--B--C--X);
label("$A$",A,SW)... |
gpt | The Angle Bisector Theorem tells us that \[\frac{BX}{AX}=\frac{BC}{AC}=\frac{27}{30}=\frac{9}{10}.\]Since $\triangle BCX$ and $\triangle ACX$ share the same height, the ratio of their areas is simply the ratio of their bases, so our answer is \[\frac{BX}{AX}=\boxed{\frac{9}{10}}.\] |
human | Let $\triangle ABC$ have side lengths $AB=13$, $AC=14$, and $BC=15$. There are two circles located inside $\angle BAC$ which are tangent to rays $\overline{AB}$, $\overline{AC}$, and segment $\overline{BC}$. Compute the distance between the centers of these two circles. |
gpt | The two circles described in the problem are shown in the diagram. The circle located inside $\triangle ABC$ is called the incircle; following convention we will label its center $I$. The other circle is known as an excircle, and we label its center $E$. To begin, we may compute the area of triangle $ABC$ using Hero... |
human | 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 an altitude of $24$ centimeters, the area of its lower base is $225\pi$ sq cm and the area of its upper base is $25\pi$ sq cm, what is the altitude of the small cone that was cut off? [asy]s... |
gpt | The two bases are circles, and the area of a circle is $\pi r^2$. If the area of the upper base (which is also the base of the small cone) is $25\pi$ sq cm, then its radius is $5$ cm, and the radius of the lower base is $15$ cm. The upper base, therefore, has a radius that is $\frac{1}{3}$ the size of the radius of t... |
human | A cube with an edge length of 4 units has the same volume as a square-based pyramid with base edge lengths of 8 units and a height of $h$ units. What is the value of $h$? |
gpt | The cube has volume $4^3=64$. The pyramid has volume $\frac{1}{3}8^2h$. So
$$64=\frac{64}{3}h\Rightarrow h=\boxed{3}$$ |
human | An isosceles, obtuse triangle has one angle with a degree measure that is 50$\%$ larger than the measure of a right angle. What is the measure, in degrees, of one of the two smallest angles in the triangle? Express your answer as a decimal to the nearest tenth. |
gpt | An angle with measure $50\%$ larger than the measure of a right angle has measure $\frac{3}{2}\cdot 90^{\circ}=135^{\circ}$.
Thus the other two angles have a combined measure of $45^{\circ}$. Each one has a measure of
$$\frac{45^{\circ}}{2}=\boxed{22.5^{\circ}}.$$ |
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