2. A circle is inscribed in a square of diagonal length 12 inches. What is the area of the circle?

Transcription

1 March 24, 2011

2 1. When a square is cut into two congruent rectangles, each has a perimeter of P feet. When the square is cut into three congruent rectangles, each has a perimeter of P 6 feet. Determine the area of the square. a. 144 ft 2 b. 324 ft 2 c. 576 ft 2 d. 900 ft 2 e ft 2 2. A circle is inscribed in a square of diagonal length 12 inches. What is the area of the circle? a. 18 sq in b. 72π sq in c. 36π sq in d. 18π sq in e. 36 sq in 3. Let be nine times and the measure of the supplement of be nine times the measure of the supplement of. Determine the. a. 8 b. 10 c. 12 d. 15 e A unit square is translated 4 units to the right and then 4 units down and then directly back to the starting position. What is the total area swept out by the traveling square? Some locations are swept more than once; only count them once in the total area. a. 16 units 2 b. 20 units 2 c. 15 units 2 d. 10 units 2 e. 18 units 2 5. Compute the maximum number of intersections of 17 straight lines. a. 136 b. 120 c. 128 d. 153 e Compute the positive difference between the sum of the interior angles and the sum of the exterior angles of a regular octagon. a b. 900 c. 720 d. 810 e The top of the Leaning Tower of Pisa is meters vertically above the ground. The top of the tower is currently displaced 3.9 meters horizontally. Based on these measurements estimate the length of the tower to the nearest hundredth of a meter. a m b m c m d m e m 1

3 8. Let F be the height of an equilateral triangle and let K be the area of the triangle. Determine 3 2 a. 2F 2 b. 3.5F 2 c. 1.5F 2 d. 0.25F 2 e. 0.5F 2 9. Two similar triangles have areas of 80 in 2 and 64 in 2. If one side of the smaller triangle is 10 inches, what is the length of the corresponding side of the larger triangle to the nearest tenth of an inch? a. Not enough information b in c inch d in e in 10. In triangle ABC the angle bisector of angle A intersects side BC in the point D. Let BC = 12 cm, BD = 8 cm, and AC = 3 cm. What is the length of side AB? a. 6 cm b. 3.5 cm c. 4 cm d. 4.5 cm e. 1.5 cm 11. Let P and Q be two concentric circles of radii 8 in and 3 in, respectively. Determine the area of the region inside circle P and outside of a square inscribed in Q to the nearest square inch. a. 165 in 2 b. 178 in 2 c. 182 in 2 d. 155 in 2 e. 183 in A solid cube has edges of length one foot. The set of points precisely one foot away from the surface of the cube encloses a solid shape. Determine the volume of this shape to the nearest cubic foot. a. 7 ft 3 b. 21 ft 3 c. 15 ft 3 d. 20 ft 3 e. 17 ft A trapezoidal water trough is 10 feet long and 2 feet deep, see diagram. The lengths of each base of the isosceles trapezoid are 2 feet and 6 feet, respectively. What is the water depth when the trough is holding 12.5 cubic feet of water? a. 1.5 ft b ft c ft d ft e. 0.5 ft 2

4 14. Eighteen segments of length 2.5 inches are joined at their endpoints to create a rectangle. What is the largest possible area of the rectangle? a. 50 in 2 b. 46 in 2 c in 2 d. 125 in 2 e. 100 in A square floor is covered with square tiles. If the total number of tiles in the two diagonals is 37, how many tiles cover the floor? a. 361 b. 324 c. 289 d. 400 e If a square has side of length S and a diagonal of length D, what is? a. 2 2 b. 1 c. 1.5 d. 2 e The measures of the angles of a triangle are represented by 3x, 2y, and 3y 3x. Which of the following can be the sum of x and y if the triangle is isosceles? a. 48 b. 72 c. 40 d. 50 e Let P and Q be externally tangent circles of radii 9 inches and 3 inches and let AB be tangent to both P and Q at points A and B, respectively. What is the area of quadrilateral APQB? a. 72 in 2 b in 2 c. 36 in 2 d in 2 e in 2 3

5 19. Cube A has a surface area that is 125% of the surface area of cube B. If the volume of cube B is x% of the volume of cube A, determine x to the nearest integer. a. 68 b. 70 c. 72 d. 75 e Determine the degree measure to the nearest minute of the central angle that has an intercepted arc measuring 15 ft in a circle of diameter 19 ft. a b c d e A target is made up of three concentric circles whose radii are in the ratio 1:2:3. What is the probability that a random shot that hits the target will hit inside the second circle but outside the innermost circle? a. b. c. d. e. 22. In right triangle RST, angle R is a right angle and RS = 2RT. Point D is chosen on side ST, and perpendicular lines are drawn to sides RS and RT, intersecting at points E and F, respectively. If REDF is a square, determine the ratio of SD to DT. a. b. 4 c. d. 2 e A point A = (1, 4) is reflected about the line y = x to the point B. Next the point B is reflected about the line y = x to a point C. What is the area of triangle ABC? a units 2 b. 16 units 2 c. 15 units 2 d. 20 units 2 e. 14 units In triangle ABC side AB is the same length as side AC. The angle bisector of angle B intersects side AC in the point D and the angle bisector of angle BDA intersects side AB in the point E. Determine the ratio of a. b. c. d. 1.5 e. 3 4

6 25. How many different 3-letter strings can be formed from the letters of GEOMETRY (no letter can be used in a given string more times than it appears in the word)? a. 336 b. 228 c. 218 d. 168 e

7 SHORT ANSWER Place the answer in the appropriate space. 66. A triangle with sides 26 cm, 28 cm, and 30 cm is constructed so that the longest and shortest sides are tangent to a circle. The third side passes through the center of the circle. Determine the radius of the circle. 67. A rectangular prism has a height of 4 in. Determine the length and width of the prism if all the edges are integers, the numerical values of the surface area and volume are the same, and the volume is minimized. 68. The center of a sphere of radius 4 feet is a distance of 2 feet from a plane intersecting the sphere. What is the area of the circle of intersection of the plane and the sphere to the nearest tenth of a square foot? 69. Side BC of an equilateral triangle ABC is extended through the point B to a point D such that the distance from D to side AC is 10 feet and the distance from D to side AB extended is 4 feet. What is the height of triangle ABC to the nearest foot? 70. Let ABGFHCDE be the vertices of the cube pictured below with sides of length 3 yards and let K be the midpoint of edge HE. Determine the length of BK to the nearest tenth of a yard. 6

8 1. b 2. d 3. e 4. c 5. a 6. c 7. a 8. e 9. b 10. a 11. e 12. b 13. e 14. d 15. a 16. d 17. a 18. b 19. c 20. d 21. e 22. e 23. c 24. c 25. d cm in by 8 in ft ft yd 7

9 1. Let 3x be the length of the side of the square. Then the conditions give and Hence x = 6 feet. This gives an area of 324 ft The side of the square is 18 inches. Hence the area is 18π square inches. 3. Let x be and y be. Then x = 9y and Solving gives Translating 4 units right and then 4 units down gives 9 units. The diagonal translation sweeps an area of 6 additional units. The diagonal sweep is a trapezoid made from an isosceles right triangle with legs length 4 units minus an isosceles right triangle with legs length 2 units The sum of the interior angles of a regular octagon is 8*135 = The sum of the exterior angles of a regular octagon is 360. Hence the difference is m 8. Let 2x be the side of the triangle. Then 9. Let x be the corresponding side. Then and. Hence 0.5 solving gives x = 11.2 inches. 10. Let E be the point on side AC such that line DE is parallel to side AB. It is clear that triangle ABC is similar to triangle EDC. Also triangle ADE is an isosceles triangle because angle BAD is congruent to angle ADE by alternate interior angles of parallel lines, but angle BAD is congruent to angle DAE by the definition of angle bisector. Using the two pieces of information it can be shown that AE = DE = 2 cm. Furthermore using similarity again it can be shown that AB = 6 cm square inches. 12. There are seven cubes of volume one cubic foot the original cube plus the cubes on each of the six faces. There are three cylinders of height one foot and radius one foot a fourth of a cylinder on each of the twelve edges. Finally there is one sphere of radius one foot and eighth of a sphere at each of the eight vertices. Adding these all up gives approximately 21 cubic feet. 13. Let x be the depth of the water. Then Solving for x gives 0.5 feet. 14. The largest possible area is a rectangle of whose sides are 4 segments by 5 segments or 10 inches by 12.5 inches. Thus the largest area is 125 square inches. 15. The number of tiles in one diagonal must be 19. Hence the total number of tiles is 19 2 = by the Pythagorean Theorem so 2. 8

10 Hence y = 36. If 3 2, then x = 24. If 3 3 3, then x = 18. If 2 3 3, then x = 12. Hence 60 or 54 or Sides PA and QB are both perpendicular to side AB. Let T be a point on side PA such that segment QT is perpendicular to segment PA. PTQ is a right triangle with sides 6 inches, 12inches, and 6 3 inches. Hence the quadrilateral APQB has an area of 36 3 square inches. 19. Let s be the side of cube A. Let r be the side of cube B. Then or 0.8. Hence this gives.7155 or 72% or 90 and 28 minutes Let x be the length of side RT, 2x be the length of side RS, and y be the side of the square REDF. Triangle SED is similar to triangle SRT, hence. Solving gives x = 1.5y. Also triangle SED is similar to triangle DFT, hence Point B is the point 4,1 and point C is the point 1, 4. The area of the triangle is 15 square units. The easiest way to see this is to draw the rectangle through 1, 4; 1,4; 4,4; and 4, 4. This rectangle has an area of 40 square units, then subtract off the area of the three extra triangles. 24. Let. Then 0.5 and 1.5. Hence 0.75 and Thus. 25. Assume all three letters chosen are different then there are 210 different strings. If the two E s are chosen then there are six ways to choose the additional letter and 3 ways to arrange the 3 letters. Hence an additional 18 strings. This gives 228 strings. 66. Let x be the length of the radius, then by Heron s Theorem the area of the triangle is or 336 square centimeters. The triangle can be split into two triangles of height x and bases of 30 and 26. Hence, so x = 12 cm. 67. Let x be the length and y be the width of the prism. Then or. Using the table feature of the calculator the solution is 8 inches by 8 inches. 68. The radius of the circle of intersection of the plane and the sphere can be found using the Pythagorean Theorem to be 12 feet, hence the area is 37.7 square feet to the nearest tenth of a square foot. 69. The length of CD is feet and the length of DB is feet. Hence the side of the triangle ABC is feet. That makes the height of the triangle 6 feet. 70. The length of CK is feet. Hence the length of BK is feet or 4.5 feet. 9