Special Segments in Triangles

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Slide 1

Special Segments in Triangles Midsegment Median Altitude Angle Bisector Perpendicular Bisector

Slide 2

Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. and

Slide 3

Examples 1) If DE = 4, find AB. 2) If DE = 3x + 7 and AB = 44. Find x. 3) If DE = x – 1 and AB = 6x – 18, then what does AB equal? So MN equals…

Slide 4

Perpendicular Bisector Theorem In a plane, if point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP is the  bisector of AB, then CA = CB Circumcenter - The point of concurrency of the three perpendicular bisectors of a triangle. If PD, PE, and PF are perpendicular bisectors, then PA = PB = PC.

Slide 5

Example: 1) BD is the perpendicular bisector of AC. Find AD. 2) In the diagram, the perpendicular bisectors of ΔABC meet at point G and are shown in blue. Find BG. 3x + 14 = 5x 14 = 2x 7 = x So, 5x = 5(7) AD = 35 BG = AG AG = 9 So, BG = 9

Slide 6

Angle Bisector Theorem If AD bisects BAC and DB  AB and DC  AC, then DB = DC Incenter - The point of concurrency of the three angle bisectors of a triangle If AP, BP,and CP are angle bisectors of  ABC then PD = PE = PF.

Slide 7

In the diagram, N is the incenter of ABC. Find ND. Examples: 3x + 5 = 4x - 6 5 = x - 6 5x = 6x - 5 -x = - 5 FN = x x2 + 162 = 202 x2 + 256 = 400 x2 = 144 x = 12 FN = 12 So, ND = 12 FN = ND x = 15 x = 11 x = 5

Slide 8

Median of a triangle - Segments whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Centroid of a triangle - The point of concurrency (intersection) of the 3 medians of a triangle. **the medians of a triangle intersect at a point that is 2/3 of the distance from each vertex to the midpoint of the opposite side.

Slide 9

Example: P is the centroid of QRS and PT = 5. Find RT and RP. Q R S T P Example: G is the centroid , BG = 27, find GF & BF.

Slide 10

Altitude of a triangle - Perpendicular segment from a vertex to the opposite side. **can lie inside, outside, or on the triangle. Orthocenter – intersection of the 3 altitudes of a triangle. Where is the orthocenter located in the following triangles? 1) Acute triangle 2) Obtuse Triangle

Slide 11

Example: P is the orthocenter of QRS, RC = 12, and RS = 15 . Find RA. Example: G is the orthocenter , EC = 20, and AE = 15. Find the perimeter of ACE. So… P = 20 + 15 + 25 P = 60 x2 + 122 = 152 x2 + 144 = 225 x2 = 81 x = 9 RA = x SA is an altitude, therefore, ARS is a right triangle. AC = x 152 + 202 = x2 225 + 400 = x2 625 = x2 25 = x

Summary: This powerpoint includes theorems, defintions, and examples for perpendicular bisectors, angle bisectors, midsegment theorem, altitudes, and medians of triangles

Tags: median altitude angle bisector perpendicular incenter orthocenter centroid circumcenter

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