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2. Bisectors, Medians, and Altitudes Inequalities and Triangles The Triangle Inequality 2 Triangles & Inequalities Indirect Proof $100 $100 $100 $100 $100 $200 $200 $200 $200 $200 $300 $300 $300 $300 $300 $400 $400 $400 $400 $400 $500 $500 $500 $500 $500

3. Bisectors, Medians, and Altitudesfor $100 Define: orthocenter

4. Answer Orthocenter –The intersection point of the altitudes of a triangle. Back

5. Bisectors, Medians, and Altitudes for $200 Where can the perpendicular bisectors of the sides of a right triangle intersect?

6. Answer On the triangle. Back

7. Bisectors, Medians, and Altitudes for $300 Where is the center of the largest circle that you could draw inside a given triangle? What is the special name for this point?

8. Answer The intersection of the angle bisectors of a triangle; the point is called the incenter. Back

9. Bisectors, Medians, and Altitudes for $400 Find the center of the circle that you can circumscribe about the triangle.

10. Answer The circumcenter is made by the perpendicular bisectors of a triangle. Only need to find the Intersection of 2 lines: Median of AB is (-3, ½) Perp Line: y = 1/2 Median of BC is (-1, ½) Perp Line: x = -1 Cicumcenter: (-1, 1/2) A B C Back

11. Bisectors, Medians, and Altitudes for $500 In triangle ACE,G is the centroid and AD = 12. Find AG and GD.

12. Answer The centroid divides the medians of a triangle into parts of length (2/3) and (1/3) so, AG = (2/3)*(AD) = (2/3)(12) = 8 GD = (1/3)*(AD) = (1/3)(12) = 4 Back

13. Inequalities and Triangles for $100 Define: Comparison Property

14. Answer For all real numbers a, b: a<b, a=b, or a>b Back

15. Inequalities and Triangles for $200 Define: Inequality

16. Answer For any real numbers a and b, a>b iff there is a positive number c such that a = b + c Back

17. Inequalities and Triangles for $300 If in triangle ABC, AB = 10, BC = 12 and CA = 9, which angle has the greatest measure?

18. Answer Angle A has the greatest measure because it is opposite side BC, which is the longest side. Back

19. Inequalities and Triangles for $400 If in triangle ABC, <A = 10 degrees, <B = 85 degrees and <C = 85 degrees, which side is the longest?

20. Answer Side AC and Side AB are the longest because they are opposite the largest angles (85 degrees). Since there are two equal angles, the triangle is isosceles. Back

21. Inequalities and Triangles for $500 Define the exterior angle inequality theorem

22. Answer If an angle is the exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles Back

23. Indirect Proof for $100 Define: Indirect Reasoning

24. Answer Indirect reasoning – reasoning that assumes the conclusion is false and then shows that this assumption leads to a contradiction. Back

25. Indirect Proof for $200 List the three steps for writing an indirect proof:

26. Answer List the three steps for writing an indirect proof: • Assume that the conclusion is false • Show that this assumption leads to a contradiction of the hypothesis, or some other fact, such as a definition, postulate, theorem, or corollary • Point out that because the false conclusion leads to an incorrect statement, the original conclusion must be true Back

27. Indirect Proof for $300 Prove that there is no greatest even integer.

28. Answer Assume that there is a greatest even integer, p. Then let p+2 = m m>p and p can be written 2x for some integer x since it is even. Then: p+2 = m; 2x+2 = m; 2(x+1) = m. x+ 1 is an integer, so 2(x+1) means m is even. Thus m is an even number and m>p Contradiction against assuming p is the greatest even number Back

29. Indirect Proof for $400 Prove that the negative of any irrational number is also irrational.

30. Answer Assume x is an irrational number, but -x is rational. Then -x can be written in the form p/q where p,q are integers and q does not equal 0,1. x = -(p/q) = -p/q : -p and q are integers and thus -p/q is a rational number Contradiction with x is irrational Back

31. Indirect Proof for $500 Given: Bobby and Kina together hit at least 30 home runs. Bobby hit 18 home runs. Prove: Kina hit at least 12 home runs.

32. Answer Assume Kina hit fewer than 12 home runs. This means Bobby and Kina combined to hit at most 29 home runs because Kina would have hit at most 11 home runs and Bobby hit 18, so 11+18 = 29. This contradicts the given information that Bobby and Kina together hit at least 30 home runs. The assumption is false. Therefore, Kina hit at least 12 home runs. Back

33. The Triangle Inequalityfor $100 Write the triangle inequality theorem:

34. Answer The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Back

35. The Triangle Inequalityfor $200 The shortest segment from a point to a line is_______

36. Answer The segement perpendicular to the line that passes through the point. Back

37. The Triangle Inequalityfor $300 Can the following lengths be sides of a triangle? 4, 5, 9

38. Answer No, 4+5 = 9, in order to be a triangle 4+5 > 9 Back

39. The Triangle Inequalityfor $400 Determine the range for the measure of the third side or a triangle give that the measures of the other two sides are 37 and 43:

40. Answer 43 – 37 = 6 43 + 37 = 80 So the range for the third side, x, is: 6 < x < 80 Back

41. The Triangle Inequalityfor $500 Prove that the perpendicular segment from a point to a line is the shortest segment from the point to the line: P 1 2 3 l A B

42. Answer Back

43. 2 Triangles & Inequalitiesfor $100 Write out the SAS Inequality theorem

44. Answer If two sides of a triangle are congruent to two sides of another triangle, and the included angle in one triangle has a greater measure than the included angle in the other, then the third side of the first triangle is longer than the third side of the second triangle. Back

45. 2 Triangles & Inequalitiesfor $200 Write out the SSS Inequality theorem

46. Answer If two sides of a triangle are congruent to two sides of another triangle, and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle. Back

47. 2 Triangles & Inequalitiesfor $300 Given: ST = PQ, SR = QR and ST = 2/3 SP Prove: m<SRP > m<PRQ Q R T P S

48. Answer Back

49. 2 Triangles & Inequalitiesfor $400 Given: KL || JH; JK = HL; m<JKH + m<HKL < m<JHK + m<KHL Prove: JH < KL K J H L

50. Answer Back