Geometry 2001 part 1 1. Point is the center of a circle with a radius of 20 inches. square is drawn with two vertices on the circle and a side containing. What is the area of the square in square inches? () 310 () 314 () 320 (D) 325 326 2. What is the maximum number of two-by-three-inch cards that will fit completely, without overlap, on the top of a ten-by-ten-inch board? () 16 () 17 () 18 (D) 19 20 3. In a rectangular room, 30 x 12 x 12, a spider (S) is on the middle of an end wall, one foot from the ceiling. fly (F) is on the end wall of the far side, in the middle and one foot above the floor, too frightened to move. What is the shortest distance that the spider must travel to get to the fly? S F () 38 ft () 40 ft () 42 ft (D) 44 ft 46 ft 4. line has equation 9y + 2x = 180. What is the slope of any line perpendicular to this line? () 2/9 () 2/9 () 9/2 (D) 9/2 undefined 5. What is the equation that represents the plane that passes through the midpoints of the three segments formed by (8, 0, 0), (0, 6, 0), and (0, 0, 10)? () 15x + 20y + 12z = 120 () 15x + 20y + 12z = 60 () 20x + 15y + 12z = 120 (D) 16x + 9y + 25z = 60 8x + 6y + 10z = 480 Geometry Examination -- Part 1 -- pril 2001 Page 1
6. In the figure shown, the circles with centers O and R each have a radius of 2 units. If PQ = 1, then what is the perimeter of rectangle KLMN? () () (D) 8 units 16 units 20 units 21 units 22 units K N O P Q R L M 7. Find the ratio of the area of Triangle I to Triangle II in this rectangle. (D) 1:1 1:2 (F) 2:1 (G) 3 2 :1 1 2 :1 D I E II 8. Three faces of a box have areas as shown. What is the volume of the box? () 33 cubic units () 1107 cubic units () 192 cubic units 24 (D) 384 cubic units None of the above. 48 32 9. circle, a square, and a triangle are drawn overlapping in the plane. What is the maximum possible number of intersection points determined by these three figures? () 15 () 16 () 18 (D) 20 25 10. Point (a, b) is reflected over the x-axis and then is reflected over the y-axis. fter these reflections, the new coordinates of the point are (c, d). What is the value of ab cd? () 0 () 1 () 1 (D) 1 2 None of these Geometry Examination -- Part 1 -- pril 2001 Page 2
11. Triangle with vertices (2, 4), (6, 4), and (4, 10) is graphed on a coordinate plane. What will be the sum of the abscissas of the coordinates of the vertices when triangle is reflected over the line x = 8? () 18 () 24 () 36 (D) 40 45 12. What is the ratio of the area of the shaded square to the area of the large square? () 1/6 () 1/7 () 1/8 (D) 1/12 1/16 13. Let PQRS be a square piece of paper. P is folded onto R, and then Q is folded onto S. The area of the resulting figure is 9 square inches. Find the perimeter of square PQRS. () 9 inches () 16 inches () 18 inches (D) 24 inches 36 inches P S Q R 14. The area of the trapezoid below is 30 square units. What is the value of x? () 2 () 4 () 8 (D) 16 32 2x - 6 x + 2 15. 4 x 4 x 4 cubical box contains sixty-four identical small cubes that exactly fill the box. How many of these small cubes touch a side or the bottom of the box? () 48 () 52 () 60 (D) 64 80 2x Geometry Examination -- Part 1 -- pril 2001 Page 3
16. What is the radius of a circle inscribed in a rhombus with diagonals of length 10 units and 24 units? () () () (D) 4 units 58/13 units 60/13 units 5 units 6 units 17. The side lengths of a triangle are 4 centimeters, 6 centimeters, and 9 centimeters. One of the side lengths of a similar triangle is 36 centimeters. What is the maximum number of centimeters possible in the perimeter of the second triangle? () 190 cm () 171 cm () 134 cm (D) 114 cm 76 cm 18. Square D has an area of 36 square meters. DE = 2E. What is the ratio of the area of triangle ED to the area of square D? E () 1 2 () 1/3 () 1:4 (D) 1/5 2:3 D 19. In the diagram below, D = 6 km, = 3 km, and DE = 5 km. What is the number of kilometers in E? D () 6 km () 7 km () 8 km (D) 9 km 10 km E 20. Ray PR is tangent to circle O at R. If the measure of angle P is 41, what is the measure of arc QR? P R Q O () 49 () 50 () 51 (D) 52 53 Geometry Examination -- Part 1 -- pril 2001 Page 4
21. large cube is dipped into red paint and then divided into 125 smaller congruent cubes. One of the smaller cubes is then randomly selected. What is the probability that the cube selected will have at least 25 percent of its surface area painted red? () 25/125 () 34/125 () 35/125 (D) 44/125 52/125 22. Find the exact value of D, given that = 24 and = 8. () 8 3 () 4 3 () 3 3 (D) 4 2 6 E D 23. trapezoid is inscribed in a circle of radius 3 cm so that one base is a diameter of the circle and the other base has length 3 cm. What is the perimeter of the trapezoid? () 9 cm () 12 cm () 15 cm (D) 18 cm 20 cm 24. ongruent isosceles triangles are cut from the corners of a square so that the remaining figure has an area equal to three-fourths of the area of the square. How long are the legs of the triangles if the square has sides of length s units? () () () S units 22 2 units S 2 S 23 units 3 2S units 3 22 units 25. If the full moon covers an angle of about 30 minutes in the sky, how many full moons placed next to each other would extend from one point on the horizon to the point on the opposite side of the horizon? () 90 () 180 () 270 (D) 360 None of these Geometry Examination -- Part 1 -- pril 2001 Page 5
26. Let m be the line with equation x = 6. Let n be the line with equation y = 2. Give the magnitude of the rotation resulting from reflecting over line m after reflecting over line n. () 90 () 180 () 270 (D) 60 None of these 27. certain type of decorative tissue paper is sold in rolls of radius 5 cm and has a cardboard core of radius 1 cm. If the paper is used until the thickness of the paper layer is reduced to half its original thickness, what fraction of the original amount of tissue paper is still on the roll? () 1/3 () 1 2 () 1/4 (D) 2/3 3 4 28. The bases of two quadrangular prisms are similar with ratio of similitude 1.5, (i.e. ratio of 3:2). The prisms have the same height. What is the ratio of their volumes? () 1.5 () 2.25 () 3 (D) 4.5 5 29. windmill is turning in the wind. If the length of a blade is 12 meters and it makes 15 revolutions in a minute, how far will a point on its tip have traveled in an hour? () 18,500 π () 20,600 π () 25,600 π (D) 21,500 π 21,600 π 30. transformation S maps (x, y) onto ( y, x). Which word best describes S? () reflection () slide () turn (D) walk flip 31. Which of the following is not a sufficient condition for a parallelogram? () () () (D) both pairs of opposite sides are congruent both pairs of opposite angles are congruent one pair of sides is parallel diagonals bisect each other None of the above Geometry Examination -- Part 1 -- pril 2001 Page 6
32. Which polygon below cannot be used as the main polygon to tesselate a plane? () () () (D) n equilateral triangle square regular pentagon regular hexagon ll of the above will tesselate a plane. 33. One leg of a right triangle is twice the length of the other. How many times larger than the smaller leg is the hypotenuse? () 5 () 3 () 7 (D) 2 2 2 34. The two legs of a right triangle have lengths 6x and 7x. What is the perimeter of the triangle? () 15x+ x 85 () 13x+ x 85 () 42x+ x 85 (D) 13x+ x 42 15x+ x 42 35. It takes you about 110 seconds to walk around a circular garden. t this rate, about how long would it take you to walk through the garden along the diameter? () 35 seconds () 29 seconds () 79seconds (D) 74 seconds 57 seconds 36. car s tire has a radius of 1 foot. bout how many revolutions does the tire make while the car goes 1 mile? () 2640 () 1680 () 1320 (D) 840 660 37. In the figure below, is at (2, y) and is at (3, 4). Find the approximate value of y for which O = O. () 3.0 () 3.5 () 4.0 (D) 4.3 4.6 O Geometry Examination -- Part 1 -- pril 2001 Page 7
38. sphere has a volume of 2304π cubic units. What is the area of a great circle of that sphere in square units? () 24π () 72π () 144π (D) 12π 48π 39. right circular cone has a height of 24 mm and a slant height of 25 mm. Find the area of the base in square mm. () 49π () 48π () 47π (D) 46π 45π 40. Find the slope of the segment connecting the origin to the midpoint of the segment with endpoints (14, -11) and ( -4, -35). () 23/5 () 23/4 () 4/3 (D) 23/5 3/4 Geometry Examination -- Part 1 -- pril 2001 Page 8
2001 Geometry part 2 1. Given: D and E intersect at = and D = DE Prove: m = m ED F E G D 2. Let X, Y, and Z represent three ships. From Ship Y, Ship X is 30 North of East and Ship Z is 10 South of east. Form Ship Z, Ship X is 15 West of North. From the captain's viewpoint on Ship X, what is the position of the other two ships? 3. Prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Draw and label a diagram and then solve. 4. One side of a triangle is double the length of a second side. The third side is triple the length of the second side. Draw a diagram of the problem and then explain why this is impossible. 5. Let = (-1, 3) and = (11, 2). Prove that the point = (3, -7) is on the circle with center and radius. Geometry Examination -- Part 2 pril 2001 Page 1
6. The distance from Point X to (2, 8) is 17. Draw a sketch of the locus of the points and then name 6 possible locations of Point X. 7. Draw a diagram and then determine the coordinates of the 12 lattice points on the circle: x 2 + y 2 = 25. 8. Show by indirect reasoning that a quadrilateral cannot have four acute angles. 9. The Earth is 4.3 light years from the star system lpha entauri, and 6.1 light years from arnard s Star, the two closest star systems to us. From this information only, how far are these systems from each other? Draw and label a diagram and then solve. 10. Find a counterexample to the conditional statement: If,, and are on the same line and = 20 and = 5, then = 25. Geometry Examination -- Part 2 pril 2001 Page 2
2001 Geometry 1. 2. 3. 4. 5. 6. E 7. 8. 9. D 10. 11. 12. 13. D 14. 15. 16. 17. 18. 19. E 20. 21. D 22. 23. 24. 25. D 26. 27. 28. 29. E 30. 31. 32. 33. 34. 35. 36. D 37. E 38. 39. 40. D