1. In the figure, A is the circumcenter of ΔLMN
In the figure, A is the circumcenter of ΔLMN. Find y if LO = 8y + 9 and ON = 12y – 11.
A. –5
B. 0.5
C. 5
D. 10
5-Minute Check 1

2. In the figure, A is the circumcenter of ΔLMN. Find x if mAPM = 7x + 13.
B. 11
C. 7
D. –13
5-Minute Check 2

3. In the figure, A is the circumcenter of ΔLMN
In the figure, A is the circumcenter of ΔLMN. Find r if AN = 4r – 8 and AM = 3(2r – 11).
A. –12.5
B. 2.5 C
D. 12.5
5-Minute Check 3

4. In the figure, point D is the incenter of ΔABC
In the figure, point D is the incenter of ΔABC. What segment is congruent to DG?
___
A. DE
B. DA
C. DC
D. DB
___
5-Minute Check 4

5. In the figure, point D is the incenter of ΔABC
In the figure, point D is the incenter of ΔABC. What angle is congruent to DCF?
A. GCD
B. DCG
C. DFB
D. ADE
5-Minute Check 5

6. Which of the following statements about the circumcenter of a triangle is false?
A. It is equidistant from the sides of the triangle.
B. It can be located outside of the triangle.
C. It is the point where the perpendicular bisectors intersect.
D. It is the center of the circumscribed circle.
5-Minute Check 6

7. Identify and use medians in triangles.
You identified and used perpendicular and angle bisectors in triangles.
Identify and use medians in triangles.
Identify and use altitudes in triangles.
Then/Now

8. Concept

9. In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.
Use the Centroid Theorem
In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.
Example 1

10. In ΔLNP, R is the centroid and LO = 30. Find LR and RO.
A. LR = 15; RO = 15
B. LR = 20; RO = 10
C. LR = 17; RO = 13
D. LR = 18; RO = 12
Example 1

11. Use the Centroid Theorem
In ΔABC, CG = 4. Find GE.
Example 2

12. In ΔJLN, JP = 16. Find PM.
A. 4
B. 6
C. 16
D. 8
Example 2

13. Find the Centroid on a Coordinate Plane
SCULPTURE An artist is designing a sculpture that balances a triangle on top of a pole. In the artist’s design on the coordinate plane, the vertices are located at (1, 4), (3, 0), and (3, 8). What are the coordinates of the point where the artist should place the pole under the triangle so that it will balance?
Understand You need to find the centroid of the triangle. This is the point at which the triangle will balance.
Example 3

14. BASEBALL A fan of a local baseball team is designing a triangular sign for the upcoming game. In his design on the coordinate plane, the vertices are located at (–3, 2), (–1, –2), and (–1, 6). What are the coordinates of the point where the fan should place the pole under the triangle so that it will balance?
A. −8 5 , 2 B.
C. (–1, 2) D. (0, 4)
Example 3

15. Concept

16. Find the Orthocenter on a Coordinate Plane
COORDINATE GEOMETRY The vertices of ΔHIJ are H(1, 2), I(–3, –3), and J(–5, 1). Find the coordinates of the orthocenter of ΔHIJ.
Example 4

17. Find an equation of the altitude from The slope of
Find the Orthocenter on a Coordinate Plane
Find an equation of the altitude from The slope of
so the slope of an altitude is
Example 4

18. Next, find an equation of the altitude from I to The
Find the Orthocenter on a Coordinate Plane
Next, find an equation of the altitude from I to The
slope of so the slope of an altitude is –6.
Example 4

19. Find the Orthocenter on a Coordinate Plane
Then, solve a system of equations to find the point of intersection of the altitudes.
Example 4

20. COORDINATE GEOMETRY The vertices of ΔABC are A(–2, 2), B(4, 4), and C(1, –2). Find the coordinates of the orthocenter of ΔABC.
A. (1, 0)
B. (0, 1)
C. (–1, 1)
D. (0, 0)
Example 4

21. Concept