1. Over Lesson 5–1 5-Minute Check 1 A.–5 B.0.5 C.5 D.10 In the figure, A is the circumcenter of ΔLMN. Find y if LO = 8y + 9 and ON = 12y – 11.

2. Over Lesson 5–1 5-Minute Check 4 In the figure, point D is the incenter of ΔABC. What segment is congruent to DG? ___ A.DE B.DA C.DC D.DB ___

3. Over Lesson 5–1 5-Minute Check 5 A.  GCD B.  DCG C.  DFB D.  ADE In the figure, point D is the incenter of ΔABC. What angle is congruent to  DCF?

4. Concept

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

6. 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.

7. 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.

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

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

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

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

12. Concept