solid geometry - Ratio

SOURCE:COMPETITION
Number of Problems: 14. : (Book)

A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter $10$ and altitude $12$ , and the axes of the cylinder and cone coincide. Find the radius of the cylinder.

$extbf{(A)} frac{8}3qquad extbf{(B)} frac{30}{11}qquad extbf{(C)} 3qquad extbf{(D)} frac{25}{8}qquad extbf{(E...$

In rectangle $ABCD, AB=5$ and $BC=3$ . Points $F$ and $G$ are on $overline{CD}$ so that $DF=1$ and $GC=2$ . Lines $AF$ and $BG$ intersect at $E$ . Find the area of $riangle AEB$ .

$extbf{(A) } 10 qquad extbf{(B) } frac{21}{2} qquad extbf{(C) } 12 qquad extbf{(D) } frac{25}{2} qquad extbf{(E) }...$

Points $A,B,C,D,E$ and $F$ lie, in that order, on $overline{AF}$ , dividing it into five segments, each of length 1. Point $G$ is not on line $AF$ . Point $H$ lies on $overline{GD}$ , and point $J$ lies on $overline{GF}$ . The line segments $overline{HC}, overline{JE},$ and $overline{AG}$ are parallel. Find $HC/JE$ .

$ext{(A)} 5/4 qquad ext{(B)} 4/3 qquad ext{(C)} 3/2 qquad ext{(D)} 5/3 qquad ext{(E)} 2$

Points $E$ and $F$ are located on square $ABCD$ so that $riangle BEF$ is equilateral. What is the ratio of the area of $riangle DEF$ to that of $riangle ABE$ ?

$mathrm{(A) } frac{4}{3} qquad mathrm{(B) } frac{3}{2} qquad mathrm{(C) } sqrt{3} qquad mathrm{(D) } 2 qqua...$

Right $riangle ABC$ has $AB=3, BC=4,$ and $AC=5.$ Square $XYZW$ is inscribed in $riangle ABC$ with $X$ and $Y$ on $overline{AC}, W$ on $overline{AB},$ and $Z$ on $overline{BC}.$ What is the side length of the square?

$extbf{(A) } frac{3}{2} qquad extbf{(B) } frac{60}{37} qquad extbf{(C) } frac{12}{7} qquad extbf{(D) } frac{23}{13...$

A circle of radius 1 is tangent to a circle of radius 2. The sides of $riangle ABC$ are tangent to the circles as shown, and the sides $overline{AB}$ and $overline{AC}$ are congruent. What is the area of $riangle ABC$ ?

size(200); pathpen = linewidth(0.7); pointpen = black;real t=2^0.5;D((0,0)--(4*t,0)--(2*t,8)--cycle);D(CR((2*t,2),2));D(CR((2...

$mathrm{(A) } frac{35}{2}qquadmathrm{(B) } 15sqrt{2}qquadmathrm{(C) } frac{64}{3}qquadmathrm{(D) } 16sqrt{2...$

''>''

$mathrm{(A) } frac{35}{2}qquadmathrm{(B) } 15sqrt{2}qquadmathrm{(C) } frac{64}{3}qquadmathrm{(D) } 16sqrt{2...$

Consider the $12$ -sided polygon $ABCDEFGHIJKL$ , as shown. Each of its sides has length $4$ , and each two consecutive sides form a right angle. Suppose that $overline{AG}$ and $overline{CH}$ meet at $M$ . What is the area of quadrilateral $ABCM$ ?

$ext{(A)} frac {44}{3}qquad ext{(B)} 16 qquad ext{(C)} frac {88}{5}qquad ext{(D)} 20 qquad ext{(E)} frac...$

Rectangle $ABCD$ has $AB=8$ and $BC=6$ . Point $M$ is the midpoint of diagonal $overline{AC}$ , and $E$ is on $AB$ with $overline{ME}perpoverline{AC}$ . What is the area of $riangle AME$ ?

$ext{(A) } frac{65}{8}qquad ext{(B) } frac{25}{3}qquad ext{(C) } 9qquad ext{(D) } frac{75}{8}qquad ext{(E) } fra...$

10.

Points $A$ and $B$ lie on a circle centered at $O$ , and $angle AOB = 60^circ$ . A second circle is internally tangent to the first and tangent to both $overline{OA}$ and $overline{OB}$ . What is the ratio of the area of the smaller circle to that of the larger circle?

$mathrm{(A)} frac{1}{16}qquadmathrm{(B)} frac{1}{9}qquadmathrm{(C)} frac{1}{8}qquadmathrm{(D)} frac{1}{6}qquad...$

11.

Trapezoid $ABCD$ has bases $overline{AB}$ and $overline{CD}$ and diagonals intersecting at $K$ . Suppose that $AB = 9$ , $DC = 12$ , and the area of $riangle AKD$ is $24$ . What is the area of trapezoid $ABCD$ ?

$mathrm{(A)} 92qquadmathrm{(B)} 94qquadmathrm{(C)} 96qquadmathrm{(D)} 98 qquadmathrm{(E)} 100$

12.

Triangle $ABC$ has a right angle at $B$ . Point $D$ is the foot of the altitude from $B$ , $AD=3$ , and $DC=4$ . What is the area of $riangle ABC$ ?

$mathrm{(A)} 4sqrt3qquadmathrm{(B)} 7sqrt3qquadmathrm{(C)} 21qquadmathrm{(D)} 14sqrt3 qquadmathrm{(E)} 42$

13.

' The area of $riangle$ $EBD$ is one third of the area of $3-4-5$ $riangle$ $ABC$ . Segment $DE$ is perpendicular to segment $AB$ . What is $BD$ ?

unitsize(10mm);defaultpen(linewidth(.8pt)+fontsize(10pt));dotfactor=4;pair A=(0,0), B=(5,0), C=(1.8,2.4), D=(5-4sqrt(3)/3,0),...

$extbf{(A)} frac{4}{3} qquad extbf{(B)} sqrt{5} qquad extbf{(C)} frac{9}{4} qquad extbf{(D)} frac{4sqrt{3}}{3...$

14.

Externally tangent circles with centers at points A and B have radii of lengths 5 and 3, respectively. A line externally tangent to both circles intersects ray AB at point C. What is BC?

$extbf{(A)} 4qquad extbf{(B)} 4.8qquad extbf{(C)} 10.2qquad extbf{(D)} 12qquad extbf{(E)} 14.4$