1. Medians and Altitudes of Triangles And Inequalities in One Triangle
Chapter 5.2 and 5.3
Medians and Altitudes of Triangles
And
Inequalities in One Triangle

2. Definitions
Median of a Triangle – a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side.
Centroid – The point of concurrency of the medians of a triangle.

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.
Centroid Theorem
YV = 12
Simplify.

5. 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

6. Use the Centroid Theorem
In ΔABC, CG = 4. Find GE.

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

8. Example 3
Find the Centroid on a Coordinate Plane
SCULPTURE Lee 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.

9. Example 3
Find the Centroid on a Coordinate Plane

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

11. Definition
Altitude of a triangle - segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side.
(aka height)

12. Concept

13. Concept

14. Concept

15. Concept

16. Concept

17. Use the Exterior Angle Inequality Theorem

18. Use the Exterior Angle Inequality Theorem

19. A. B. C. D.

20. A. B. C. D.

21. Concept

22. List the angles of ΔABC in order from smallest to largest.
Order Triangle Angle Measures
List the angles of ΔABC in order from smallest to largest.
Answer: C, A, B

23. List the angles of ΔTVX in order from smallest to largest.
A. X, T, V
B. X, V, T
C. V, T, X
D. T, V, X

24. List the sides of ΔABC in order from shortest to longest.
Order Triangle Side Lengths
List the sides of ΔABC in order from shortest to longest.
Answer: AC, AB, BC

25. List the sides of ΔRST in order from shortest to longest.
A. RS, RT, ST
B. RT, RS, ST
C. ST, RS, RT
D. RS, ST, RT