solid geometry - Area of a rectangle, a triangle

SOURCE:COMPETITION
Number of Problems: 26. : (Book)

10.

11.

12.

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 $frac{a}{b}$ ?

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

13.

A unit square is rotated $45^circ$ about its center. What is the area of the region swept out by the interior of the square?

$extbf{(A)} 1 - frac{sqrt2}{2} + frac{pi}{4}qquad extbf{(B)} frac{1}{2} + frac{pi}{4} qquad extbf{(C)} 2 - sq...$

14.

A regular octagon $ABCDEFGH$ has sides of length two. Find the area of $riangle ADG$ .

$extbf{(A) } 4 + 2sqrt2 qquad extbf{(B) } 6 + sqrt2qquad extbf{(C) } 4 + 3sqrt2 qquad extbf{(D) } 3 + 4sqrt2 q...$

15.

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

$ext{(A)} pi qquad ext{(B)} 1.5pi qquad ext{(C)} 2pi qquad ext{(D)} 3pi qquad ext{(E)} 3.5pi$

16.

Equilateral $riangle ABC$ has side length $2$ , $M$ is the midpoint of $overline{AC}$ , and $C$ is the midpoint of $overline{BD}$ . What is the area of $riangle CDM$ ? $extrm{(A)} frac {sqrt {2}}{2}qquad extrm{(B)} frac {3}{4}qquad extrm{(C)} frac {sqrt {3}}{2}qquad extrm{(D...$

17.

In the overlapping triangles $riangle{ABC}$ and $riangle{ABE}$ sharing common side $AB$ , $angle{EAB}$ and $angle{ABC}$ are right angles, $AB=4$ , $BC=6$ , $AE=8$ , and $overline{AC}$ and $overline{BE}$ intersect at $D$ . What is the difference between the areas of $riangle{ADE}$ and $riangle{BDC}$ ?

$mathrm {(A)} 2 qquad mathrm {(B)} 4 qquad mathrm {(C)} 5 qquad mathrm {(D)} 8 qquad mathrm {(E)} 9 qquad$

'''>'In the overlapping triangles $riangle{ABC}$ and $riangle{ABE}$ sharing common side $AB$ , $angle{EAB}$ and $angle{ABC}$ are right angles, $AB=4$ , $BC=6$ , $AE=8$ , and $overline{AC}$ and $overline{BE}$ intersect at $D$ . What is the difference between the areas of $riangle{ADE}$ and $riangle{BDC}$ ?

$mathrm {(A)} 2 qquad mathrm {(B)} 4 qquad mathrm {(C)} 5 qquad mathrm {(D)} 8 qquad mathrm {(E)} 9 qquad$

18.

A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle? $mathrm{(A) } 20piqquad mathrm{(B) } 25piqquad mathrm{(C) } 30piqquad mathrm{(D) } 40piqquad mathrm{(E) ...$

19.

A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

$mathrm{(A) } 120 qquad mathrm{(B) } 180 qquad mathrm{(C) } 240 qquad mathrm{(D) } 360 qquad mathrm{(E) } 4...$

20.

Rhombus $ABCD$ is similar to rhombus $BFDE$ . The area of rhombus $ABCD$ is $24$ and $angle BAD = 60^circ$ . What is the area of rhombus $BFDE$ ?

$mathrm{(A) } 6qquad mathrm{(B) } 4sqrt{3}qquad mathrm{(C) } 8qquad mathrm{(D) } 9qquad mathrm{(E) } 6sqr...$

21.

An equiangular octagon has four sides of length 1 and four sides of length $frac{sqrt{2}}{2}$ , arranged so that no two consecutive sides have the same length. What is the area of the octagon?

$mathrm{(A) } frac72qquad mathrm{(B) } frac{7sqrt2}{2}qquad mathrm{(C) } frac{5+4sqrt2}{2}qquad mathrm{(D)...$

22.

In $ABC$ we have $AB = 25$ , $BC = 39$ , and $AC=42$ . Points $D$ and $E$ are on $AB$ and $AC$ respectively, with $AD = 19$ and $AE = 14$ . What is the ratio of the area of triangle $ADE$ to the area of the quadrilateral $BCED$ ?