1997 Geometry part 1

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1 1997 Geometry part 1 1. In the figure below is similar to E, is the midpoint of segment and the perimeter of E = 10. Find the perimeter of. a) 15 b) 0 c) 5 d) 30 e) 40 E. In the figure below m 1 = m and m 3 = m 4. What line(s) is(are) parallel to segment? a) F b) E c) G d) oth F and E e) no lines parallel 1 F 3 E G 4 3. The area of a rectangular region is equal to the area of a square region. If the measures of the two adjacent sides of the rectangle are 4a and 8a, find the length of the side of the square. a) 4a b) 4a c) 6a d) 8a e) 8a 4. Sphere S has center S and radius of length x. Sphere T has center T and radius of length y. If ST = x + y, then the intersection of sphere S and sphere T is. a) a line b) a point c) a plane d) a circle e) does not exists 5. If M (-,-1) is the midpoint of segment, where has coordinates (6,3), find the coordinates of. a) (-10,-5) b) (-8,-3) c) (-5,-) d) (4, ) e) (-4, -1.5) 6. etermine the type of quadrilateral given (0,0), (0,10), (6,18) (6,8). (I) parallelogram (II) trapezoid (III) square (IV) rhombus a) I only b) II only c) IV only d) I and IV only e) I, III, and IV only 1997 Geometry part 1

2 7. In proving the theorem, If the measures of two minor arcs are equal, then the measures of their central angles are equal, the student drew the diagram below and selected the statements for his proof of the theorem. Study the statements and decide which answer below best describes them. Statements 1. m 1 = m. m = m 3. m 1 = m His list of statements... a) is sufficient for the proof. b) is not arranged in logical order. c) does not contain the hypothesis of the theorem. d) does not contain the conclusion of the theorem. e) is missing both the hypothesis and the conclusion of the theorem. 8. In QRS, QR RS, m Q = 45, and QR = 8. Find QS. 8 a) 4 b) 4 c) d) 8 e) 8 9. The vertices of have coordinates (-1,-1), (,) and (-5,4). The median from intersects at point. Find the coordinates of. a) (-3, 1.5) b) (-1.5, 3) c) (0.5, 0.5) d) (1.5, 1.5) e) (1, 1) 10. Find the equation for the area of the figure below in terms of a, b, c, and d. a) bd + ac - ad d b) bc - ad c) bd + ad - ac d) bd + ac e) none of the above b 11. Quadrilateral is inscribed in a circle, m = x, m = x, and m = x +0. Find the measure of. a) 0 b) 80 c) 100 d) 160 e) In parallelogram, E = E, F is the midpoint of, G EF, G =, and = 8. Find the area of EF. c O a a) 4 b) 8 c) 8 3 d) 1 e) 16 E G F 1997 Geometry part 1

3 13. If a woman walks 3 miles east, then 6 miles north, and 5 miles east, how far will she be from the starting point? a) miles b) 6 miles c) 8 miles d) 10 miles e) 14 miles 14. What is the x-coordinate of the point on the x-axis that is equidistant from the points (-,0) and (0,4)? a) -10 b) 7 c) 0 d) 3 e) In the figure, how many units are in the perimeter of if ll? Express your answer in simplest radical form. a) 4 b) 41 c) d) e) ,, = 8, = 3 and = 1. How many units are in the perimeter of? a) 3 b) 36 c) 44 d) 48 e) How many square inches are in the area of the regular octagon with center P as shown? Express your answer as a decimal to the nearest hundredth. a) b) 9.4 c) d) 153. e) " P 18. What is the number of units in the circumference of the circle with center at P (-,3) and passing through Q (10,-)? Express your answer in terms of π. a) 10π b) 13π c) 16.π d) π e) 6π 1997 Geometry part 1

4 19. The volume of a given sphere is 36π cubic inches. How many square inches are in its surface area? Express your answer in terms of π. a) 9π b) 1π c) 36π d) 54 π e) 79 π 0. One base of a given isosceles trapezoid is half the length of the other, each of the congruent sides are 5 inches long, and the area of the trapezoid is 36 square inches. What is the sum of the least pair of integers that can be used as measures of the base? a) 9 b) 1 c) 15 d) 18 e) 1 1. The difference between the perimeters of two squares is 0 units and the sum of the perimeters is 5 units. What is the ratio of the area of the smaller square to the area of the larger square? a) 4 to 9 b) 9 to 16 c) 16 to 5 d) 16 to 81 e) 5 to 81. Each side of a triangle has a length that is a different even integer number of units. What is the least possible number of units in its perimeter? a) 1 b) 14 c) 16 d) 18 e) 0 3. How many square inches are in the total area of the shaded triangles given that is a trapezoid with bases measuring 1 and 18 and height measuring 6? a) 18 b) 36 c) 54 d) 7 e) water tower in a small South akota town is shaped like the one shown in the diagram. Use the information given in the figure to find the number of cubic feet in volume of the tank. Express your answer in terms of π. a) 564 π 8 ft b) 600 π c) 67 π 6 ft d) 756 π 6 ft e) 148π 5. How many square inches are in the area of the trapezoid inscribed in the circle with center O as shown? Express your answer as a decimal to the nearest tenth. a) 41.6 b) 83.1 c) 96 d) e) 10 8" O 1997 Geometry part 1 30

5 6. rectangle with a length-to-width ratio of 1+ 5 is called a golden rectangle. If the perimeter of a golden rectangle is 100 cm, what is the approximate number of square centimeters in the area of the rectangle? a) 16 b) 477 c) 590 d) 361 e) The outer figure in the diagram is a rectangle. How many square units are in the area of the shaded polygon? Express your answer as a decimal, rounded to the nearest tenth. a) b) 51.1 c) 4.13 d) 40 e) is a square with M, N, O and P the midpoints of segments,, and, respectively. What is the number of square inches in the area of MNOP when = 16 inches? a) 3 b) 64 c) 6 d) 18 e) n isosceles right triangle with legs of length 8 is partitioned into sixteen congruent triangles as shown. The shaded area is which of the following? 4 6 a) 10 b) 0 c) 3 d) 40 e) What is the maximum number of points of intersection when two circles and three straight lines intersect each other? ssume that no figure coincides with another. a) 5 b) 11 c) 13 d) 15 e) Sammy wants to put a brick border around a tree. The border is to be placed 1.5 meters from the tree. If the circumference of the tree is 56.5 cm, what is the inner circumference of the brick border? (Use 3.14 for pi.) a) 5.95 b) c) 9.4 d) 9.99 e) Find a right triangle with integral side lengths and area 84 square units. 8 a) 1, 14, 15 b) 8,1,5 c) 10, 17, 1 d) 7, 4, 5 e) none possible 1997 Geometry part 1

6 33. If = = 3E = 4 and point x is randomly selected on segment, what is the probability that x is between E and? Give your answer as a common fraction. E a) 1/3 b) 1/4 c) /5 d) 1/6 e) 3/8 34. Three cubes of volume 1, 8, and 7 are glued together at their faces. What is the smallest possible surface area of the resulting configuration? a) 36 b) 56 c) 70 d) 7 e) Find the area of the shaded region. a) 1 b) 9 c) 8 d) 6 e) If the strip of triangles shown is folded to form an octahedron and each vertex is assigned the value of the sum of the four triangular faces to which it belongs, find the maximum value of a vertex. a) b) 3 c) 4 d) 5 e) If is a square and E = y, the area of E is 5 E a) b) 4 y 5 y c) 3y d) e) Geometry part 1

7 38. Ray bisects and lies on segment. If = 6, =14, and = 14, find. a) 6 b) 8.4 c) 9.8 d) 7 e) In RST, line SU is the perpendicular bisector of segment RT and U lies on segment RT. Which statement(s) must be true? (I) RST is equilateral (II) RSU is congruent to TSU (III) ray SU is the bisector of RST a) I only b) II only c) III only d) II and III only e) I, II,and III 40. is equiangular. What is b? a) 60 b) 65 c) 70 d) 75 e) 80 b a a a a 1997 Geometry part 1

8 1997 Geometry part 1. Find the shortest segment in the figure F 58 E. Given circumscribed polygon where = 1, = 15, and = 5, find. 3. is equilateral with X = X, YZ, and m 1 = 105. If YZ = 1, find XY. 1 Y X Z 4. Given the cube as shown, show that F is congruent to G. H G E F 1997 Geometry part

9 5. parallelogram has vertices whose coordinates are (0,0), (,3) and (5,0) in a rectangular coordinate system. The fourth vertex can be located at any one of three possible points. How many square units are in the area of the triangle formed by these three possible points? 6. Find FE. 1 E 1 F 1 7. bamboo eighteen units high was broken by the wind. Its top touched the ground six units from the root. Find the lengths of the segments of the bamboo. 8. Prove: If R is any acute angle, (sin R) + (cos R) = 1. (Hint: From any point on one side of R, draw a perpendicular to the other side) 1997 Geometry part

10 9. Sketch the image of the indicated glide reflection: vector (0,-3) and line y-axis Geometry part

11 1997 Geometry nswer Key E E E E E

12 1998 Geometry part 1 1. If two regular polygons have the same number of sides, the polygons must a. be congruent. b. be similar. c. have congruent apothems. d. have equal perimeters. e. None of these.. block of metal with dimensions 5" by 6" is melted down and made into wire with a diameter of 0.4". Find the approximate wire length. a. 00 ft b. 50 ft c. 0 ft d. 10 ft e. 5 ft 3. Two concentric squares are shown in Figure 1. If a point is selected at random from the interior of the larger square, what is the probability that it is in the shaded region? 10 cm 8 cm 9 9 a. b. c P (-4, 9) is rotated about the origin. Find P '. d. 1 5 e. 4 5 a. ( 4,9) b. ( 4,9) c. ( 4, 9 ) d. ( 4, 9) e. None of these 5. In an isosceles triangle, the length of one of the legs is always a. equal to the length of the base. b. equal to one-half the length of the base. c. greater than one-half the length of the base. d. less than one-half the length of the base. e. annot be determined. 6. The measure of the angle of a sector is 10 0 and the area of the sector is 7. What is the length of the radius of the circle? a. 3 3 b. 6 3 c. 9 d. 18 e. None of these 7. In Figure, is scalene, and angles and are complementary. If all the angles are positive, then angle must be?. Figure a. median of. b. bisector of. c. altitude of. d. perpendicular bisector of. e. None of these Geometry part 1

13 8. ngles and are supplementary, and angles and are complementary. If all the angles are positive, then angle must be?. a. acute b. right c. obtuse d. straight e. annot be determined 9. If is a parallelogram, then which statement is always true? a. b. c. and bisect each other d. and bisect the angles through which each pass e. None of these 10. If the segment is drawn connecting the midpoints of and in Figure 3, then that segment has a slope of: y (7, 10) (11, 6) (5. 3) x a. 1 4 b. 1 3 c. 1 d. e. None of these 11. It is possible to pass a plane through a cube in such a way that the intersection is I. an equilateral triangle II. a trapezoid III. a pentagon IV. a hexagon a. I only b. II only c. I, II, and IV d. I, II, III, and IV e. None of these 1. If > and EF > GH, then which statement is not always true? 1 a. > 1 b. EF > GH c. +EF > + GH d. EF > GH e. ll are true. 13. Givne the statement: "ll similar triangles are congruent." Which assertion is correct regarding the given statement? a. The statement is true but its converse is false. b. The statement is false but its converse is true. c. The statement and its converse are both false. d. The statement and the negative of the statement are both false. e. None of these. 14. The area of a circle inscribed in a regular hexagon is 100. The area of the hexagon is?. a. 600 b. 300 c. 00 d e Geometry part 1

14 15. In Figure 4 were accurately drawn, which segment would be shortest? Figure 4 (not drawn to scale) a. F b. F c. E d. FG e. E 16. If the number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area, fined the length of an edge expressed in feet. a. 6 b. 864 c. 178 d. 10,368 e If successive midpoints of a quadrilateral are joined, the new quadrilateral must be a?. a. rectangle b. square c. rhombus d. parallelogram e. None of these. 18. Grandma rented a rowboat and took off from the dock as shown in Figure 5. She decided to row the boat on a course such that her lines of sight back to the dock and her favorite fishing spot on shore were always perpendicular to each other. In what type of path on the lake did she travel? ock oat X Fishing Spot a. line b. elliptical c. circular d. annot be determined. e. None of these. 19. RSTU is a trapezoid with m R = 150 o, m U = 135 o,ru = 9, and RS = 18. Find the perimeter of RSTU. a b c d e. None of these 0. triangle has sides of length r, r, and s, where r, s >0. Which of the following are possible? I. r = 1 s II. r = s III. r = s a. I only b. II only c. I and II d. I and III e. I, II, and III 1998 Geometry part 1

15 1. In Figure 6, is a right triangle, is between and, and sin = 3. Find the tangent of angle 5. Figure 6 a. 5 b. 3 4 c. 4 5 d. 4 3 e. None of these. circle with center O has radius r. segment is tangent to the circle at and has length 4 3 r. What is the shortest distance from to the circle? 5 a. 3 r b. 5 6 r c r d. 8 r e. None of these 3 3. If the radius of a circle is increased by 100%, the area is increased by?. a. 100% b. 00% c. 300% d. 400% e. None of these 4. is a rhombus shown in Figure 7. Which of the following is not necessarily true? 3 5 Figure 7 a. m 5 = 90 o b. m = m 3 c. m = m 4 d. m 5 > m 4 e. ll are true foot ladder is placed against a vertical wall of a building. The foot of the ladder is 7 feet from the base of the building. If the top of the ladder slips 4 feet, then the foot of the ladder will slide?. a. 9 ft b. 15 ft c. 5 ft d. 8 ft e. 4 ft 6. In, = 4 cm, = 10 cm, = 6 cm. The radius of the inscribed circle is a. 6 cm b. 4 cm c. 13 cm d. 8 cm e. None of these. 7. The length of rectangle is 5 inches and its width is 3 inches. iagonal is divided into three equal segments by E and F. The area of EF (in square inches) is?. 4 a. 3 b. 5 3 c. 5 d e Geometry part 1

16 8. is the hypotenuse of a right triangle. Median = 7, and median E = 4. Find the length of. a. 10 units b. 5 3 units c. 5 units d. 13 units e. 15 units 9. If two circles have exactly three common tangents, then the circles must be?. a. concentric b. non-intersecting c. internally tangent d. externally tangent e. eccentric 30. In Figure 8, quadrilateral is a rectangle. XY, XZ, K, XJ K, = 156, = 65, and K = 60. XY + XZ =? X N Z J K M Figure 8 a. 55 b. 60 c. 65 d. annot be determined e. None of these. 31. The ratio of the lengths of the bases of two parallelograms is :3 and the ratio of the corresponding angles is 3:. What is the ratio of the areas of the parallelograms? a. 3: b. :3 c. 4:9 d. 9:4 e. None of these. 3. The area of a ring between to concentric circles is 1 1 square millimeters. The length of a chord of the larger circle tangent to the smaller circle, in millimeters, is?. 5 a. b. 5 c. 5 d. 10 e The number of points equidistant from a circle and two parallel tangents to the circle is?. a. 0 b. c. 3 d. 4 e. infinite 34. Suppose a balloon is blown up at a steady rate. It takes 5 seconds for the radius of the balloon to become 5 inches. How many more seconds will it take for the radius of the balloon to become 10 inches? a. 5 seconds b. 10 seconds c. 0 seconds d. 40 seconds e. None of these. 35. In shown in Figure 9, point R divides in the ratio of 1:. Let S be the point of intersection of sides and T where T is the midpoint of R. Find the ratio S:S. R Y T S Figure 9 a. 1:4 b. 1:3 c. :5 d. 4:11 e. 3: Geometry part 1

17 36. The area of the largest triangle that can be inscribed in a semi-circle whose radius is r is?. a. r b. r 3 c. r 3 d. 3r 3 e. 1 r 37. In Figure 10, is an equilateral triangle of side length 1. The arcs,,and are arcs of circles centered at,, and, respectively. The area of the unshaded region is?. Figure 10 a square units b. 3 4 square units c d. 6 4 square units e. square units square units 38. In the triangle shown in Figure 11, and m m = 30 o. Find m. Figure 11 a. 30 o b..5 o c. 0 o d. 15 o e. 10 o 39. The sides of triangle are in the ratio :3:4. is the angle bisector drawn to the shortest side, dividing it into segments and. If = 10, then the length of the longer segment of is a. 3 1 b. 5 c d. 6 e The median of a trapezoid is 4 cm long. How many integral value combinations are possible for the measures of the bases? a. 11 b. 1 c. 3 d. 4 e. infinitely many 1998 Geometry part 1

18 1998 Geometry part 1. is a diameter of a circle. Tangents Z and Y are drawn so that Y and Z intersect in a point on the circle. If Z = a and Z = b, a b, find the diameter of the circle.. Figure 1 shows three circles, each tangent to the other two and each with a radius of. The three circles are bound together with a tightly wrapped string. Find the length of the string. Figure 1 3. convex polygon is bounded by the x-axis, x = 1, x=4, and y =mx + 4. If its area is 7, find m. 4. Find the area of the triangle in Figure. P L cm Figure 5. In, the medians and N to sides and, respectively, intersect in point O. P is the midpoint of side, and MP intersects N in Q. If the area of MQ is 5, find the area of Geometry part

19 6. The pitch of a typical roof in Las Vegas is 4:1; that is 1' 4' What length "x4" (rafter plus ft.) is required for the following roof line (Figure 3) if we want a -foot overhang? Give your answer to the nearest inch. ft. rafter 36 ft. Figure 3 7. etermine the cost of tiles required to tile a rectangular patio 0 feet wide by 36 feet long if a circular fountain (no tiles required) 6 feet in diameter is to be installed in the center of the patio. The 8-inch by 8-inch tiles cost $.90 each. ssume you need to order 1% more tiles because of the waste in cutting t8iles to fit around the fountain. 8. shed is 8 feet long, 8 feet wide, and 8 feet high. hungry spider is at the middle of an end wall, 1 foot above the floor. It sees a meal in the form of a frightened fly at the middle of the other end wall, 1 foot from the ceiling. Find the shortest distance the spider would have to crawl to reach the fly. 9. regular hexagonal pyramid with base edge 6 and height 8 is inscribed in a cone. Find the total lateral area of the pyramid and cone. 10. If UK is tangent to circle L and UE = LU, present a logical argument to explain why UE bisects LK. (See Figure 4 ) L E U K 1998 Geometry part

20 1998 Geometry nswer Key E E E E E

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