Medians and Altitudes of Triangles LESSON 5–2 Medians and Altitudes of Triangles
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
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
In the figure, A is the circumcenter of ΔLMN. Find x if mAPM = 7x + 13. B. 11 C. 7 D. –13 5-Minute Check 2
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. 10.25 D. 12.5 5-Minute Check 3
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
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
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
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
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
median centroid altitude orthocenter Vocabulary
Concept
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
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
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
Use the Centroid Theorem In ΔABC, CG = 4. Find GE. Example 2
Use the Centroid Theorem CG = 4 6 = CE Example 2
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
In ΔJLN, JP = 16. Find PM. A. 4 B. 6 C. 16 D. 8 Example 2
Concept
Medians and Altitudes of Triangles LESSON 5–2 Medians and Altitudes of Triangles