
Special Segments in Triangles Midsegment Median Altitude Angle Bisector Perpendicular Bisector
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
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…
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.
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
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.
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
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.
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.
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
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
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