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Geometry Review 1. Which of the following is not true about the medians of a triangle? a. Regardless of the triangle’s shape, the three medians always meet at a single point. b. The point where the medians intersect is called an incenter. c. The point where the medians intersect is called a centroid. Name ____________________________________ A# ________ 2. Which of the following is unnecessary to prove that the 2 base angles of an isosceles triangle are congruent? a. The angle sum theorem for triangles b. SAS postulate c. The definition of an angle bisector d. The definition of congruent triangles d. A triangle has three medians. 3. Which of the following is necessary to prove that a triangle’s exterior angle equals the sum of the two remote interior angles? a. The definition of complementary angles b. The definition of an angle bisector 4. Similar triangles are triangles that are the same shape but not the same size. How can we determine whether triangles are similar or not? a. The ratio between each corresponding side is the same. c. The definition of supplementary angles b. One would lie directly on top of the other and match precisely. d. The definition of congruent triangles c. All three angles are equal. d. Both a. and c. are correct. 5. A segment connects the midpoints of two sides of a triangle. What is true about this segment? a. It is always horizontal and half the length of each side. b. It is always perpendicular to the two sides it joins and forms an isosceles triangle with the portions of the sides above it. c. It is always parallel to the third side and half as long as the third side. d. It is always half the length of those two sides and parallel to the third. 6. Assuming that ̅̅̅̅ 𝐴𝐵 ∥ ̅̅̅̅ 𝐷𝐶 and that the two triangles in the figure are similar, which triangle is similar to ∆𝐴𝐵𝐸? D a. ∆𝐶𝐷𝐸 E A C b. ∆𝐷𝐸𝐶 c. ∆𝐸𝐷𝐶 B d. All of the above 7. Mike has to build two similar tents for his buddy Jim. Both tents have a 70° angle at the top. He asks Jim for the blueprints, and Jim gives him the diagrams below. What are the side lengths of the larger tent? D A 6 ft B 70° 7.47 ft 70° y 6.96ft C 8. In the diagram below, what is the measure of ∠𝐸 if the two triangles are similar? D z A F 11.21 ft B E a. 6 ft, 6.96 ft, and 7.47 ft b. 8 ft, 9.96 ft, and 11.21 ft c. 9 ft, 10.44 ft, and 11.21 ft d. 10 ft, 11.21 ft, and 12.63 ft ̅̅̅̅ into 9. A line parallel to a triangle’s side splits 𝐴𝐵 ̅̅̅̅ lengths 12 and 5. The other side, 𝐴𝐶 , is split into lengths of x and 10. What is the length of ̅̅̅̅ 𝐴𝐶 ? 30° C F E a. 30° b. 60° c. 70° d. 90° 10. The hypotenuse of a right triangle has length 13 units, and one leg has length 12 units. How long is the other leg? 11. Given EFGH is a square with a diagonal drawn 12. What is the measure of a central angle with an from ∠𝐸 to ∠𝐺. Complete the proof that ∆𝐸𝐹𝐺 ≅ arc measure of 45°? ∆𝐺𝐻𝐸. ̅̅̅̅ and 𝐹𝐺 ̅̅̅̅ ≅ 𝐻𝐸 ̅̅̅̅ ≅ 𝐺𝐻 ̅̅̅̅ true? Why? a. Is 𝐸𝐹 b. Is ̅̅̅̅ 𝐸𝐺 ≅ ̅̅̅̅ 𝐸𝐺 true? Why? c. Is ∆𝐸𝐹𝐺 ≅ ∆𝐺𝐻𝐸 true? Why? 13. What is the measure of an inscribed angle with arc measurement of 70°? 14. What is the measure of the arc with an inscribed angle of 18°? 15. What is the measure of the major arc formed by 16. What is the measure of the angle formed two tangents that are perpendicular to one between a tangent and the diameter of a circle? another? 17. Given an arc length of 20 ft and 𝜃 = 90°, find the radius of the circle. 𝜋 19. Given a radius with length 4.5 in and 𝜃 = 6 , find the area of the sector. 18. Given an arc length of 43.2 m and a radius of 21. 4 m, find 𝜃. 20. What is the equation of the circle with center at (4, -3) and radius of 2? a. (𝑥 − 4)2 + (𝑦 + 3)2 = 2 b. (𝑥 − 4)2 + (𝑦 + 3)2 = 4 c. (𝑥 + 4)2 + (𝑦 − 3)2 = 4 d. (𝑥 − 4)2 − (𝑦 − 3)2 = 4 21. What are the center and radius of the circle described by the equation 𝑥 2 + 𝑦 2 − 18𝑥 + 12𝑦 + 68 = 0? 22. What is the equation for the circle centered on W ̅̅̅̅̅? with radius 𝑊𝑍 a. Center (6, -9); radius 7 W b. Center (-9, 6); radius 7 c. Center (-6, 9); radius 7 d. Center (9, -6); radius 7 Z a. (𝑥 − 2)2 + (𝑦 − 4)2 = √125 b. (𝑥 − 2)2 + (𝑦 − 4)2 = 125 c. (𝑥 + 3)2 + (𝑦 + 6)2 = √125 d. (𝑥 + 3)2 + (𝑦 + 6)2 = 125 23. Obviously distracted by Peeta’s witty charm, 24. What is the equation of the parabola with focus Katniss’s shot missed by a mile while her (-4, 9) and directrix y = 6? arrow followed the trajectory of the 1 4 61 parabola with focus (75, 125) and directrix a. 𝑦 = 6 𝑥 2 − 3 𝑥 + 6 y = 150. Which equation describes the arrow’s trajectory? 1 4 61 b. 𝑦 = − 6 𝑥 2 + 3 𝑥 − 6 1 4 c. 𝑦 = 6 𝑥 2 + 3 𝑥 + 1 9 61 6 61 d. 𝑥 = − 20 𝑦 2 − 10 𝑦 − 20 Write the equation in vertex-form: ___________________________________________________ Choose the correct standard-form equation. 1 a. 𝑦 = − 50 𝑥 2 + 3𝑥 + 25 1 b. 𝑦 = 50 𝑥 2 − 3𝑥 − 25 1 c. 𝑦 = − 50 𝑥 2 + 3𝑥 + 875 1 d. 𝑦 = 50 𝑥 2 + 3𝑥 + 25 25. A triangle has a perimeter of 100 26. Shmikea, the newly imported Norwegian centimeters and one side is 35 centimeters. superstore sells beautiful 6-foot round shag The other two sides have a ratio of 5:8. What rugs. Borg’s living room is a 12 by 18 foot is the length of the longest side of the rectangle, and his goal is to cover as much of the triangle? floor as possible with these rugs without them overlapping. How much of the floor space will be a. 65 centimeters left uncovered if he goes through with his plan? (Use the approximation 𝜋 = 3.14.) b. 40 centimeters a. 7.74 ft2 c. 35 centimeters b. 46.44 ft2 d. 5 centimeters c. 169.56 ft2 d. 216 ft2