1. Medians and Altitudes of Triangles
LESSON 5–2
Medians and Altitudes of Triangles

2. Five-Minute Check (over Lesson 5–1) TEKS 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
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
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
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
___
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
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.
5-Minute Check 6

9. G.6(D) Verify theorems about the relationships in
Targeted TEKS
G.6(D) Verify theorems about the relationships in
triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.
Mathematical Processes
G.1(F), G.1(G)
TEKS

10. 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

11. median
centroid
altitude
orthocenter
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. 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

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

17. Use the Centroid Theorem
CG = 4
6 = CE
Example 2

18. Subtract 4 from each side.
Use the Centroid Theorem
CG + GE = CE
Segment Addition
4 + GE = 6
Substitution
GE = 2
Subtract 4 from each side.
Answer: GE = 2
Example 2

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

20. Concept

21. Medians and Altitudes of Triangles
LESSON 5–2
Medians and Altitudes of Triangles