C Challenge

Use the figure below for Exercises 29 and 30.

29. Proof

    Given: Δ A B C cap delta eh b c  is scalene, m ∠ A B X = 36 , m ∠ C B X = 36 m angle eh b x , equals 36 , comma m angle c b x , equals 36

    Prove: X B ¯ x b bar  is not perpendicular to A C ¯ . eh c bar , .

30. Proof

    Given: Δ A B C cap delta eh b c  is scalene, m ∠ A B X = 36 , m ∠ C B X = 36 m angle eh b x , equals 36 , comma m angle c b x , equals 36

    Prove: A X ¯ ≇ X C ¯ eh x bar , neither approximately nor actually equal to , x c bar

Standardized Test Prep

SAT/ACT

31. What temporary assumption is the first step of the following indirect proof?

    Given: The sides of Δ S F K cap delta s f k  measure 3 cm, 4 cm, and 5 cm.

    Prove: The orthocenter of Δ S F K cap delta s f k  is on the triangle.

    1. The incenter of Δ S F K cap delta s f k  is inside the triangle.
    2. The orthocenter of Δ S F K cap delta s f k  is inside the triangle.
    3. The centroid of Δ S F K cap delta s f k  is outside the triangle.
    4. The orthocenter of Δ S F K cap delta s f k  is inside or outside the triangle.
32. Δ L M N ≅ Δ O P Q , m ∠ L = 39 , cap delta l m n approximately equal to cap delta o p q comma m angle , l equals 39 , comma  and m ∠ P = 61 . m angle , p equals 61 , .  What is m ∠ Q ? m angle q question mark
    6. 39
    7. 80
    8. 100
    9. 141
33. In the diagram, what are the coordinates of the circumcenter of Δ K M F ? cap delta k m f question mark

    

    1. (1.5, 1.5)
    2. (6, 0)
    3. (3, 1.5)
    4. (0, 0)

Short Response

34. For what types of triangles are the centroid, circumcenter, incenter, and orthocenter all inside the triangle? Explain.

Mixed Review

See Lesson 5-4.

35. The distances from the centroid of a triangle to its vertices are 16 cm, 17 cm, and 18 cm. What is the length of the shortest median?
36. The orthocenter of isosceles Δ A B C cap delta eh b c  lies outside the triangle and m ∠ A = 30 . m angle , eh equals 30 , .  What are the measures of the two other angles?

See Lesson 2-4.

37. You think, “If I leave home at 7:10, I'll catch the 7:25 bus. If I catch the 7:25 bus, I'll get to school before class starts. I am leaving and it's 7:10, so I'll get to school on time.” Which law of deductive reasoning are you using?

Get Ready! To prepare for Lesson 5-6, do Exercises 38–40.

See Problem 1-7.

Graph Δ A B C . cap delta eh b c .  List the sides in order from shortest to longest.

38. A(5, 0), B(0, 8), C(0, 0)
39. A ( 2 , 4 ) , B ( − 5 , 1 ) , C ( 0 , 0 ) eh open 2 comma 4 close comma b open negative 5 comma 1 close comma c open 0 comma 0 close
40. A(3, 0), B(4, 3), C(8, 0)

Table of Contents