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. 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?
Analyze You need to find the centroid of the triangle. This is the point at which the triangle will balance.
Example 3

21. Determine Graph the triangle and label the vertices A, B, and C.
Find the Centroid on a Coordinate Plane
Formulate Graph and label the triangle with vertices at (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.
Determine Graph the triangle and label the vertices A, B, and C.
Example 3

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

25. Answer: The artist should place the pole at the point
Find the Centroid on a Coordinate Plane
Answer: The artist should place the pole at the point
Justify Check the distance of the centroid from point D(3, 4). The centroid should be (2) or units to the left of D. So, the coordinates of the centroid is __ 1 3 2
Example 3

26. Find the Centroid on a Coordinate Plane
Evaluate: Any side could have been used to find the centroid. Using all three sides is a simple way to check the solution. Point P appears to be in the middle of the triangle, so the answer is reasonable.
Example 3

27. 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. (–1, 2) D. (0, 4)
Example 3

28. Concept

29. 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

30. 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
Point-slope form
Distributive Property
Add 1 to each side.
Example 4

31. 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.
Point-slope form
Distributive Property
Subtract 3 from each side.
Example 4

32. Equation of altitude from J
Find the Orthocenter on a Coordinate Plane
Then, solve a system of equations to find the point of intersection of the altitudes.
Equation of altitude from J
Substitution,
Multiply each side by 5.
Add 105 to each side.
Add 4x to each side.
Divide each side by –26.
Example 4

33. Replace x with in one of the equations to find the y-coordinate.
Find the Orthocenter on a Coordinate Plane
Replace x with in one of the equations to find the y-coordinate.
Rename as improper fractions.
Multiply and simplify.
Example 4

34. Answer: The coordinates of the orthocenter of ΔHIJ are
Find the Orthocenter on a Coordinate Plane
Answer: The coordinates of the orthocenter of ΔHIJ are
Example 4

35. 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

36. Concept

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