1. Splash Screen

2. the midpoint of the opposite side.
A median of a triangle is a segment with endpoints being a vertex of a triangle and
the midpoint of the opposite side.
Every triangle has three medians that are concurrent. The point of concurrency of the
medians of a triangle is called the centroid and is always inside the triangle.
Concept

3. Concept

4. 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

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

6. 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?
Example 3

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

8. interior, exterior, or on the side of a triangle.
An altitude of a triangle is a segment from a vertex to the line containing the opposite
side and perpendicular to the line containing that side. An altitude can lie in the
interior, exterior, or on the side of a triangle.
Concept

9. 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

10. Concept

11. End of the Lesson

12. Five-Minute Check (over Lesson 5–1) CCSS 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. G.CO.10 Prove theorems about triangles.
Content Standards
G.CO.10 Prove theorems about triangles.
G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Mathematical Practices
6 Attend to precision.
3 Construct viable arguments and critique the reasoning of others.
CCSS

20. 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

21. median
centroid
altitude
orthocenter
Vocabulary