* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Honors Geometry
Survey
Document related concepts
Steinitz's theorem wikipedia , lookup
Golden ratio wikipedia , lookup
Noether's theorem wikipedia , lookup
Euler angles wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Reuleaux triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euclidean geometry wikipedia , lookup
Integer triangle wikipedia , lookup
Transcript
Honors Geometry Review Name: Hour: Chapter 5 Answers Show all work. Complete this review without a calculator. Temporary Assumption: ABC with BC > AC and A B. 1. Write an indirect proof. Given: ΔABC with BC > AC Prove: A B Argument: Since A B, by the Converse of the Base Angles Theorem, AC BC and AC = BC (definition of congruent segments). But it is given that BC > AC. A B Therefore, in ABC, if BC > AC, A B C 2. Write an indirect proof. Given: ΔABC is scalene, mABX = 36°, mCBX = 36° Prove: XB AC B A X Temporary Assumption: ABC is scalene, mABX = 36°, mCBX = 36°, XB AC Argument: Since XB AC, AXB and CXB are right angles by the definition of perpendicular segments. So, AXB CXB by the Right Angles Congruence Theorem. By the Reflexive Property of Congruence, XB XB, so ΔABX ΔCBX by ASA . By CPCTC, AB CB. Therefore, by the definition of isosceles triangle, ΔABC is isosceles. But, it is given that ΔABC is scalene. C Therefore, if ABC is scalene, mABX = 36°, mCBX = 36°, then XB AC 3. Perform the given construction. a. Construct the circle containing the three vertices Construct the circumcenter (point of concurrency of the perpendicular bisectors of the sides of the triangle) then construct the circle through the 3 points. c. Construct the incenter of the triangle shown. Construct the point of concurrency of the angle bisectors of the triangle. b. Construct an Inscribed circle in the triangle First construct the incenter (point of concurrency of the angle bisectors) Next, from that center point, construct the perpendicular through that center to one of the sides in order to get the radius of the circle. Then construct the inscribed circle. d. Construct the orthocenter of the triangle shown. Construct the point of concurrency of the altitudes of the triangle. 4. Construct the centroid of the triangle shown. Construct the point of concurrency of the medians of the triangle. Construct the perpendicular bisectors of the sides in order to find the midpoint of each side. With a straightedge, draw the segment connecting each vertex to the midpoint of the opposite side. 5. List all of the properties of a centroid of a triangle. • Centroid is the point of concurrency of the medians of a triangle • Centroid must be inside of the triangle • Centroid is 2/3 length of the median (measured from the vertex) 6. List all of the properties of the incenter of a triangle. • Incenter is the point of concurrency of the angle bisectors of a triangle • Incenter must be inside of the triangle and is equidistant from the sides of the triangle • Incenter is the center of the inscribed circle in a triangle 7. Two sides of a triangle are 7.2 and 3.5. What are the possible lengths of the third side? 3.7 < x < 10.7 8. Can 4, 4, and 8 be lengths of sides of a triangle? If so, tell why. If not, tell why not. Be very specific. No. By the Triangle Inequality Theorem, the sum of any two sides of a triangle must be greater than the length of the third side. In this triangle, 4 + 4 = 8, which is not greater than 8. 9. Which segment is the longest? Why? OA is the longest segment. A B C 8° 9° 10 ° O In ΔCOD, CD < OD < OC In ΔBOC, BC < OC < OB In ΔAOC, AB < OB < OA D 10. List the angles in order from largest to smallest. Explain your answer. 3, 1, 2 98 cm 1 2 97 cm 3 m1 = 45°. In the right triangle on the right, m3 > m2 (3 is opposite the longer leg). If m2 is greater than (or equal to) 45°, then m3 would have to be smaller than (or equal to) m2 (their sum has to be 90°), which would contradict the fact that m3 > m2. So it must be true that m2 < 45°. You should be able to complete the following problem without a calculator. 11. Consider ΔABC shown. Solve each part algebraically. Show all steps. Leave fractions in simplest form (no decimals). a. Find the coordinates of the circumcenter. C (−5, 6) B (5, 2) 1 27 − , 4 8 A (−1, −2) b. Find the coordinates of the centroid. − 1 , 2 3 c. Find the coordinates of the orthocenter. − 1 3 , − 2 4 continue B 12. Prove the Concurrency of Medians of a Triangle Theorem GIVEN: ΔABC with centroid P, and midsegment M N . 2 PROVE: A P A M 3 3 M P 2 4 1 A C N Statements Reasons 1. ΔABC with centroid P, midsegment MN 1. Given 2. MN // AB 2. Midsegment Theorem (triangles) 3. 1 2 3. Alternate Interior Angles Theorem 4. 3 4 4. Vertical Angles Congruence Theorem 5. ΔMNP ~ ΔABP 5. AA ~ 6. MP = MN 6. Definition of similar triangles AP AB 1 7. MN = • AB 2 8. MN = 1 AB 2 MP = 1 9. AP 2 7. Midsegment Theorem 8. Division Property of Equality and simplify 9. Transitive Property of Equality (steps 6, 8) 10. MP + 1 = 1 + 1 AP 2 10. Addition Property of Equality 11. MP + AP= 3 AP 2 11. Simplify 12. AP + MP = AM 12. Segment Addition Postulate AM 3 13. AP = 2 13. Substitution (steps 11, 12) 14. 2AM = 3 AP 14. Cross Products Property 15. AP = 15. Division Property of Equality and simplify 2 • AM 3 not drawn to scale C 13. Prove the Concurrency of Angle Bisectors of a Triangle Theorem 5 6 GIVEN: ΔABC, A D bisects CAB, BD bisects CBA, C D bisects ACB G DE A B, DF BC, DG C A 3 D PROVE: DG = DE = DF A Statements Reasons 1. ΔABC, AD bisects CAB, BD bisects CBA, 1. Given 2 1 F 7 4 8 • 11 12 9 10 E CD bisects ACB, DE AB, DF BC, DG CA 2. 1 2 2. Definition of angle bisector 3. 3 and 12 are right angles 3. Definition of perpendicular segments 4. 3 12 4. Right Angles Congruence Theorem 5. AD AD 5. Reflexive Property of Congruence 6. ΔADE ΔADG 6. AAS 7. DG DE 7. CPCTC 8. DG = DE 8. Definition of congruent segments 9. 9 10 9. Definition of angle bisector 10. 8 and 11 are right angles 10. Definition of perpendicular segments 11. 8 11 11. Right Angles Congruence Theorem 12. BD BD 12. Reflexive Property of Congruence 13. ΔDEB ΔDFB 13. AAS 14. DE DF 14. CPCTC 15. DE = DF 15. Definition of congruent segments 16. DG = DE = DF 16. Transitive Property of Equality (steps 8, 15) B