1. Splash Screen

2. Five-Minute Check (over Lesson 5–1) Then/Now New Vocabulary
Theorem 5.7: Centroid Theorem
Example 1: Use the Centroid Theorem
Example 2: Use the Centroid Theorem
Example 3: Real-World Example: Find the Centroid on a Coordinate Plane
Key Concept: Orthocenter
Example 4: Find the Orthocenter on a Coordinate Plane
Concept Summary: Special Segments and Points in Triangles
Lesson Menu

3. 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 A B C D
5-Minute Check 1

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

5. 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 A B C D
5-Minute Check 3

6. 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
___ A B C D
5-Minute Check 4

7. 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 A B C D
5-Minute Check 5

8. 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. A B C D
5-Minute Check 6

9. Identify and use medians in triangles.
You identified and used perpendicular and angle bisectors in triangles. (Lesson 5–1)
Identify and use medians in triangles.
Identify and use altitudes in triangles.
Then/Now

10. Content Standards G-CO.9 Prove theorems about lines and angles.
G-CO.10 Prove theorems about triangles.
G-MG.3 Apply geometric concepts in modeling situations.
Mathematical Practices
1 Make sense of problems and persevere in solving them
2 Reason abstractly and quantitatively.
6 Attend to precision.
Then/Now

11. Centroid – the intersection of triangle medians
Median – a segment that connects one vertex of a triangle to the midpoint of its opposite side
Centroid – the intersection of triangle medians
Altitude – a segment from one triangle vertex to the opposite side, perpendicular to that side
Vocabulary

12. Concept

13. 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.
Centroid Theorem
YV = 12
Simplify.
Example 1

14. YP + PV = YV Segment Addition 8 + PV = 12 YP = 8
Use the Centroid Theorem
YP + PV = YV Segment Addition
8 + PV = 12 YP = 8
PV = 4 Subtract 8 from each side.
Answer: YP = 8; PV = 4
Example 1

15. A B C D 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 A B C D
Example 1

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

17. Segment Addition and Substitution
Use the Centroid Theorem
Centroid Theorem
Segment Addition and Substitution
CG = 4
Distributive Property
Example 2

18. Subtract GE from each side. 1 3
Use the Centroid Theorem
Subtract GE from each side. __ 1 3
Answer: GE = 2
Example 2

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

20. 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

21. Find the Centroid on a Coordinate Plane
Plan Graph and label the triangle with vertices (1, 4), (3, 0), and (3, 8). Use the Midpoint Theorem to find the midpoint of one of the sides of the triangle. The centroid is two-thirds the distance from the opposite vertex to that midpoint.
Solve Graph ΔABC.
Example 3

22. Find the midpoint D of side BC.
Find the Centroid on a Coordinate Plane
Find the midpoint D of side BC.
Graph point D.
Example 3

23. Find the Centroid on a Coordinate Plane
Notice that is a horizontal line. The distance from D(3, 4 ) to A(1, 4) is 3 – 1 or 2 units.
Example 3

24. Find the Centroid on a Coordinate Plane
The centroid is the distance. So, the centroid is (2) or units to the right of A. The coordinates are
Example 3

25. 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 B C D
A. (– , 2)
B. (– , 2)
C. (–1, 2)
D. (0, 4) __ 7 3 5
Example 3

26. Concept

27. Concept

28. Homework p 338 5, 7, 10, 16-19, 24