See Problem 3.

Coordinate Geometry Find the coordinates of the orthocenter of Δ A B C cap delta eh b c .

14. A(0, 0)

    B(4, 0)

    C(4, 2)

15. A(2, 6)

    B(8, 6)

    C(6, 2)

16. A ( 0 , − 2 ) eh open 0 comma negative 2 close

    B ( 4 , − 2 ) b open 4 comma negative 2 close

    C ( − 2 , − 8 ) c open negative 2 comma negative 8 close

B Apply

Name the centroid.

Name the orthocenter of Δ X Y Z . cap delta x y z .

19. 
20. 
21. Think About a Plan In the diagram below, Q S ¯ q s bar  and P T ¯ p t bar  are altitudes and m ∠ R = 55 . m angle , r equals 55 , .  What is m ∠ P O Q ? m angle p o q question mark

    

    • What does it mean for a segment to be an altitude?
    • What do you know about the sum of the angle measures in a triangle?
    • How do you sketch overlapping triangles separately?

Constructions Draw a triangle that fits the given description. Then construct the centroid and the orthocenter.

22. acute scalene triangle, Δ L M N cap delta l m n
23. obtuse isosceles triangle, Δ R S T cap delta r s t

In Exercises 24–27, name each segment.

24. a median in Δ A B C cap delta eh b c
25. an altitude in Δ A B C cap delta eh b c
26. a median in Δ B D C cap delta b d c
27. an altitude in Δ A O C cap delta eh o c
28. Reasoning A centroid separates a median into two segments. What is the ratio of the length of the shorter segment to the length of the longer segment?

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