This ppt includes the following topics:- Medians Centroid Angle Bisector Incentre Altitude Orthocentre Perpendicular Bisector and many more.
1. Centers of Triangles or Points of Concurrency Prepared for Ms. Pullo’s Geometry Classes
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3. Example 1 In MNP, MC and ND are medians. M D P C What is NC if NP = 18? MC bisects NP…so 18/2 9 If DP = 7.5, find MP. 7.5 + 7.5 = 15
4. How many medians does a triangle have?
5. The medians of a triangle are concurrent. The intersection of the medians is called the CENTRIOD.
6. Theorem The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint.
7. Example 2 In ABC, AN, BP, and CM are medians. If EM = 3, find C EC = N P 2(3) E EC = 6 B M A
8. Example 3 In ABC, AN, BP, and CM are medians. If EN = 12, find AE = C AN = AE + EN 2(12)=24 N P E AN = 24 + B 12 AN = A M
9. Example 4 In ABC, AN, BP, and CM are medians. If EM = 3x + 4 and CE = 8x, C what is x? N P E x=4 M B A
10. Example 5 In ABC, AN, BP, and CM are medians. If CM = 24 what is CE? C CE = 2/3CM N CE = P E 2/3(24) B CE = 16 M A
11. Angle Bisector
12. Example 1 In WYZ, ZX bisects WZY . If m1 = 55, find mWZY . W mWZY 55 55 mWZY 110 X 1 2 Z Y
13. Example 2 In FHI, IG is an angle bisector. Find mHIG. F 5( x 1) I G (4 x 1) 5(x – 1) = 4x + 1 H 5x – 5 = 4x + 1 x=6
14. How many angle bisectors does a triangle have? three The angle bisectors of a triangle are concurrent ____________. The intersection of the angle bisectors is called the ________. Incenter
15. The incenter is the same distance from the sides of the triangle. Point P is called B the __________. Incenter D F P A E C
16. Example 4 The angle bisectors of triangle ABC meet at point L. • What segments are congruent? LF, DL, EL • Find AL and FL. Triangle ADL is a A right triangle, so use FL = 6 Pythagorean thm 8 D AL2 = 82 + 62 F AL2 = 100 L AL = 10 6 C E B
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18. Tell whether each red segment is an altitude of the The altitude is the “true height” of the triangle.
19. How many altitudes does a triangle have? The altitudes of a triangle are concurrent. The intersection of the altitudes is called the ORTHOCENTER.
20. Perpendicular Bisector
21. Example 1: Tell whether each red segment is a perpendicular bisector of the triangle.
22. Example 2: Find x 3x + 4 5x - 10
23. How many perpendicular bisectors does a triangle have? The perpendicular bisectors of a triangle are concurrent. The intersection of the perpendicular bisectors is called the CIRCUMCENTER.
24. The Circumcenter is equidistant from the vertices of the triangle. B PA = PB = PC P A C
25. Example 3: The perpendicular bisectors of triangle ABC meet at point P. • Find DA. DA = 6 • Find BA. BA = 12 • Find PC. PC = 10 • Use the Pythagorean Theorem B to find DP. DP2 + 62 = 102 6 D 10 DP + 36 = 100 2 DP2 = 64 P A C DP = 8
26. Tell if the red segment is an altitude, perpendicular bisector, both, or neither? NEITHER ALTITUDE PER. BOTH BISECTOR
27. IN A NUT SHELL Median – Centroid Angle Bisector – Incenter Altitude – Orthocenter Perpendicular Bisector - Circumcenter Angle Bisector: The Incentor is equidistance to the sides Perpendicular Bisector – the Circumcenter is equidistance to the vertex
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