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5.2 Bisectors of a Triangle Advanced Geometry
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PERPENDICULAR BISECTOR OF A TRIANGLE A perpendicular bisector of a triangle is a line (ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.
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3 Perpendicular Bisectors in every triangle!
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VOCAB Concurrent lines – Three or more lines (rays or segments) that intersect at the same point. Point of Concurrency – The point of intersection of concurrent lines. Circumcenter of the triangle – The point of concurrency of the perpendicular bisectors of a triangle.
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Point of Concurrency Circumcenter of the Triangle The circumcenter of the triangle can be inside the triangle, on the triangle, or outside the triangle.
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CONCURRENCY OF PERPENDICULAR BISECTORS OF A TRIANGLE The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
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Where to meet?
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Angle Bisector of a Triangle An angle bisector of a triangle is a bisector of an angle of the triangle.
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INCENTER OF THE TRIANGLE The point of concurrency of the angle bisectors is called the incenter of the triangle.
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Incenter always lies inside the triangle!
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CONCURRENCY OF ANGLE BISECTORS OF A TRIANGLE The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangles
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Example 1.What segments are congruent? 2.If AY = 15 and AP = 17, then find PY and PX.
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How to remember points of concurrency? ABI loves to eat Peanut Butter Cups. Peanut Butter Cups: Perpendicular Bisectors – Circumcenter ABI: Angle Bisectors – Incenter
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Remember! Circumcenter is equidistant to the verticies! – Can be inside, on, or outside the triangle Incenter is equidistant to the sides! – Is always inside the triangle
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