5-3 Bisectors in Triangles

Quick Review

When three or more lines intersect in one point, they are concurrent.

• The point of concurrency of the perpendicular bisectors of a triangle is the circumcenter of the triangle.
• The point of concurrency of the angle bisectors of a triangle is the incenter of the triangle.

Example

Identify the incenter of the triangle.

The incenter of a triangle is the point of concurrency of the angle bisectors. M R ¯ m r bar  and L Q ¯ l q bar  are angle bisectors that intersect at Z. So, Z is the incenter.

Exercises

Find the coordinates of the circumcenter of Δ D E F . cap delta d e f .

14. D ( 6 , 0 ) , E ( 0 , 6 ) , F ( − 6 , 0 ) d open 6 comma 0 close comma e open 0 comma 6 close comma f open negative 6 comma 0 close
15. D ( 0 , 0 ) , E ( 6 , 0 ) , F ( 0 , 4 ) d open 0 comma 0 close comma e open 6 comma 0 close comma f open 0 comma 4 close
16. D ( 5 , − 1 ) , E ( − 1 , 3 ) , F ( 3 , − 1 ) d open 5 comma negative 1 close comma e open negative 1 comma 3 close comma f open 3 comma negative 1 close
17. D ( 2 , 3 ) , E ( 8 , 3 ) , F ( 8 , − 1 ) d open 2 comma 3 close comma e open 8 comma 3 close comma f open 8 comma negative 1 close

P is the incenter of Δ X Y Z . cap delta x y z .  Find the indicated angle measure.

18. m ∠ P X Y m angle p x y
19. m ∠ X Y Z m angle x y z
20. m ∠ P Z X m angle p z x

5-4 Medians and Altitudes

Quick Review

A median of a triangle is a segment from a vertex to the midpoint of the opposite side. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.

• The point of concurrency of the medians of a triangle is the centroid of the triangle. The centroid is two thirds the distance from each vertex to the midpoint of the opposite side.
• The point of concurrency of the altitudes of a triangle is the orthocenter of the triangle.

Example

If P B = 6 , p b equals 6 comma  what is SB?

S is the centroid because A Q ¯ eh q bar  and C R ¯ c r bar  are medians. So, S B = 2 3 P B = 2 3 ( 6 ) = 4 . s b equals . 2 thirds . p b equals . 2 thirds , open 6 close equals 4 . .

Exercises

Determine whether A B ¯ eh b bar  is a median, an altitude, or neither. Explain.

21. 
22. 

23. Δ P Q R cap delta p q r  has medians Q M ¯ q m bar  and P N ¯ p n bar  that intersect at Z.

    If Z M = 4 , z m equals 4 , comma  find QZ and QM.

Δ A B C cap delta eh b c  has vertices A ( 2 , 3 ) , B ( − 4 , − 3 ) ,  and  C ( 2 , − 3 ) . eh open 2 comma 3 close comma b open negative 4 comma negative 3 close comma , and c open 2 comma negative 3 close .  Find the coordinates of each point of concurrency.

24. centroid
25. orthocenter

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