solid geometry - Length, Pythagoras theorem

SOURCE:COMPETITION
Number of Problems: 21. : (Book)

In trapezoid $ABCD$ , $overline{AB}$ and $overline{CD}$ are perpendicular to $overline{AD}$ , with $AB+CD=BC$ , $AB<CD$ , and $AD=7$ . What is $ABcdot CD$ ?

$extbf{(A)} 12 qquad extbf{(B)} 12.25 qquad extbf{(C)} 12.5 qquad extbf{(D)} 12.75 qquad extbf{(E)} 13$

10.

In $riangle ABC$ , we have $AC=BC=7$ and $AB=2$ . Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$ . What is $BD$ ?

$mathrm{(A)} 3 qquadmathrm{(B)} 2sqrt{3} qquadmathrm{(C)} 4 qquadmathrm{(D)} 5 qquadmathrm{(...$

11.

Square $ABCD$ has side length $2$ . A semicircle with diameter $overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $overline{AD}$ at $E$ . What is the length of $overline{CE}$ ?

$mathrm{(A) } frac{2+sqrt{5}}{2} qquad mathrm{(B) } sqrt{5} qquad mathrm{(C) } sqrt{6} qquad mathrm{(D) } ...$

1 Problem
2 Solution
- 2.1 Solution 1
- 2.2 Solution 2
3 See also

12.

A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?

$mathrm{(A) } frac{1}{4}x^2qquad mathrm{(B) } frac{2}{5}x^2qquad mathrm{(C) } frac{1}{2}x^2qquad mathrm{(D) ...$

13.

A circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. What is $r$ ?

unitsize(3mm);defaultpen(linewidth(.8pt)+fontsize(7pt));dotfactor=4;real r1=1, r2=1+sqrt(2);pair A=(0,0), B=(1+sqrt(2),1+sqrt...

$extbf{(A) } sqrt{2} qquad extbf{(B) } 1+sqrt{2} qquad extbf{(C) } sqrt{6} qquad extbf{(D) } 3 qquad extbf{(E) } ...$

14.

A triangle with side lengths in the ratio $3 : 4 : 5$ is inscribed in a circle with radius 3. What is the area of the triangle?

$mathrm{(A)} 8.64qquad mathrm{(B)} 12qquad mathrm{(C)} 5piqquad mathrm{(D)} 17.28qquad mathrm{(E)} 18$

''>''

A triangle with side lengths in the ratio $3 : 4 : 5$ is inscribed in a circle with radius 3. What is the area of the triangle?

$mathrm{(A)} 8.64qquad mathrm{(B)} 12qquad mathrm{(C)} 5piqquad mathrm{(D)} 17.28qquad mathrm{(E)} 18$

15.

Points $A$ and $C$ lie on a circle centered at $O$ , each of $overline{BA}$ and $overline{BC}$ are tangent to the circle, and $riangle ABC$ is equilateral. The circle intersects $overline{BO}$ at $D$ . What is $frac{BD}{BO}$ ?

$ext{(A) } frac {sqrt2}{3}qquad ext{(B) } frac {1}{2}qquad ext{(C) } frac {sqrt3}{3}qquad ext{(D) } frac {sqrt2...$

16.

Triangle $ABC$ has a right angle at $B$ , $AB=1$ , and $BC=2$ . The bisector of $angle BAC$ meets $overline{BC}$ at $D$ . What is $BD$ ?

$ext{(A) } frac {sqrt3 - 1}{2}qquad ext{(B) } frac {sqrt5 - 1}{2}qquad ext{(C) } frac {sqrt5 + 1}{2}qquad ext{(D...$

17.

A right triangle has perimeter $32$ and area $20$ . What is the length of its hypotenuse?

$mathrm{(A)} frac{57}{4}qquadmathrm{(B)} frac{59}{4}qquadmathrm{(C)} frac{61}{4}qquadmathrm{(D)} frac{63}{4}qq...$

1 Problem
2 Solution
3 See also

18.

19.

Nondegenerate $riangle ABC$ has integer side lengths, $overline{BD}$ is an angle bisector, $AD = 3$ , and $DC=8$ . What is the smallest possible value of the perimeter?

$extbf{(A)} 30 qquad extbf{(B)} 33 qquad extbf{(C)} 35 qquad extbf{(D)} 36 qquad extbf{(E)} 37$

20.

A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?

$extbf{(A)} 4+4sqrt{2} qquad extbf{(B)} 2+4sqrt{2}+2sqrt{3} qquad extbf{(C)} 2+3sqrt{2}+3sqrt{3} qquad extb...$

21.

Point B is due east of point A. Point C is due north of point B. The distance between points A and C is $10sqrt 2$ , and $angle BAC= 45^circ$ . Point D is 20 meters due north of point C. The distance AD is between which two integers?

$extbf{(A)} 30 ext{and} 31 qquad extbf{(B)} 31 ext{and} 32 qquad extbf{(C)} 32 ext{and} 33 qquad extbf{...$

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