Geometry Quarter Exam Name: ____________________________ Student Number: ____________________ Answer the questions below. Make sure to show your work and provide complete geometric explanations for full credit. 1. A right triangle has a hypotenuse length of 60, and one side length of 24. Do the side lengths form a Pythagorean triple? Explain. 2. A 16-foot ladder is placed against the side of a building as shown in Figure 1 below. The bottom of the ladder is 8 feet from the base of the building. In order to increase the reach of the ladder against the building, the ladder is moved 4 feet closer to the base of the building, as shown in Figure 2. To the nearest foot, how much farther up the building does the ladder now reach? Show how you arrived at your answer. 3. A ship travels due west for 60 miles. It then travels in a northwest direction for 85 miles and ends up 119 miles from its original position. To the nearest tenth of a degree, how many degrees north of west did it turn when it changed direction? Show your work. Geometry Quarter Exam 85 mi 119 mi 60 mi 4. Sketch two triangles (You do not have to provide the sketch). Label the lengths of the sides of Triangle A as 3, 4, and 5. Label the lengths of the sides of Triangle B as 5, 12, and 13. a. What is the sum of the measures of the acute angles of any right triangle? Explain your reasoning. b. Write the tangent ratios for the acute angles of Triangle A. c. Write the tangent ratios for the acute angles of Triangle B. d. Write a rule describing the relationship between the tangents of the acute angles of any right triangle. 5. A farmer is estimating the surface area of his barn to find how much paint he needs to buy. One part of the barn is triangular as shown. 29 29 25 m The darkened sides in the figure are the edges of the roof. This trim will be painted white. Find the length of each of these two sides of the triangle. Explain how you found the answer. Geometry Quarter Exam 6. A local park is adding a new 14 ft long slide to the playground. The vertical height of the slide is h. The distance from the bottom of the stairs to the bottom of the slide needs to be 15 ft. Also, due to safety reasons, the slide must form a 30 angle with the ground, as shown below. h x 30 a. What is the height h of the slide? Round your answer to the nearest tenth. b. To the nearest degree, what angle x will the stairs form with the ground? 7. In the diagram, the dashed-lined figure is the image of the solid-lined figure after a rigid motion. A B N J E M D C K L a. List all pairs of corresponding sides. b. Name the image of point . 8.. Suppose 9. The endpoints of and . Is m || n? Explain. are A(0, 0) and B(6, 8). a. Describe a transformation of that results in the image whose endpoints are b. What are the endpoints of the image of ? c. What quadrant does 10. after a dilation whose scale factor is and . and whose center is lie in? has vertices , , and . What are the vertices of the resulting image when the triangle is translated 4 units to the left then dilated by a scale factor of 2? Geometry Quarter Exam 11. Use this triangle. 7 in. a. b. c. d. Find the length of the hypotenuse using the special relationships for a 45 -45 -90 triangle. Explain the relationships. Write the exact value for sin 45 as a fraction in simplest form. Use the formula for the area of a triangle given SAS to find the area of the triangle. Explain your steps. Use a different method to find the area of the triangle. Explain your steps. 12. The diameter of a basketball rim is 18 inches. A standard basketball has a circumference of 30 inches. About how much room is there between the ball and the rim in a shot in which the ball goes in exactly in the center of the rim? Show your work. 13. A pie tin with a 9-inch diameter and 1.5-inch depth has a capacity of 53 fluid ounces. A giant pumpkin pie is baked every year for a fall festival, using a similarly shaped pie tin with dimensions 12 times as large. a. What is the capacity of the giant pumpkin pie tin? b. One gallon is equal to 128 fluid ounces. What is the capacity of the giant pie tin in gallons? 14. In , NL = NM, and the perimeter is 52 cm. A, B, and C are points of tangency to the circle. MC = 6 cm. Find NL. Explain your reasoning. (The figure is not drawn to scale.) C N M B A L Geometry Quarter Exam 15. a. Find x. (The figure is not drawn to scale.) b. Is the triangle equilateral, isosceles, or scalene? Explain. P (6x) (8x - 10) R Q (10x + 10) 16. Given: m X = 98, , m Y = 92. Find each measurement. (The figure is not drawn to scale.) Y X Z W a. m Z b. m c. m W d. m 17. The diameter of a circle has endpoints P(-7, -10) and Q(3, 2). a. Find the center of the circle. b. Find the radius. If your answer is not an integer, express it in radical form. Geometry Quarter Exam c. Write an equation for the circle. 18. If ABCD is a rectangle inscribed in circle O, do both diagonals contain the center of the circle? Explain. A B O D C 19. Show that it is not possible for the lengths of the segments of two intersecting chords to be four consecutive integers. m +2 m m +3 m +1 20. In a random survey of Valentine's Day candy bags, 50% of bags contained red candy, 25% of bags contained pink candy, and 15% contained both red and pink candy. Explain which is greater: -The conditional probability that a bag with pink candy also contains red candy -The conditional probability that a bag with red candy also contains pink candy