1. Splash Screen

2. Lesson Menu 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

3. Then/Now You identified and used perpendicular and angle bisectors in triangles. Identify and use medians in triangles. Identify and use altitudes in triangles.

4. Vocabulary median centroid altitude orthocenter

5. Concept

6. Example 1 Use the Centroid Theorem In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Centroid Theorem YV = 12 Simplify.

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

8. Example 1 A.LR = 15; RO = 15 B.LR = 20; RO = 10 C.LR = 17; RO = 13 D.LR = 18; RO = 12 In ΔLNP, R is the centroid and LO = 30. Find LR and RO.

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

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

11. Example 2 Use the Centroid Theorem Answer: GE = 2 Segment AdditionCG + GE = CE Substitution4 + GE = 6 Subtract 4 from each side.GE = 2

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

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

14. Example 3 Find the Centroid on a Coordinate Plane SolveGraph the triangle and label the vertices A, B, and C. PlanGraph 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.

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

16. Example 3 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.

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

18. Example 3 Find the Centroid on a Coordinate Plane Answer: The artist should place the pole at the point CheckCheck 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 3

19. Example 3 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)

20. Concept

21. Example 4 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.

22. Example 4 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.

23. Example 4 Find the Orthocenter on a Coordinate Plane Point-slope form Distributive Property Subtract 3 from each side. Next, find an equation of the altitude from I to The slope of so the slope of an altitude is –6.

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

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

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

27. Example 4 A.(1, 0) B.(0, 1) C.(–1, 1) D.(0, 0) 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.

28. Concept

29. End of the Lesson