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5-2 Bisectors of Triangles Objectives Prove and apply properties of perpendicular bisectors of a triangle. Prove and apply properties of angle bisectors of a triangle.Holt McDougal Geometry
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5-2 Bisectors of Triangles Vocabulary concurrent point of concurrency circumcenter of a triangle circumscribed incenter of a triangle inscribedHolt McDougal Geometry
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5-2 Bisectors of Triangles Helpful Hint The perpendicular bisector of a side of a triangle does not always pass through the opposite vertex.Holt McDougal Geometry
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5-2 Bisectors of Triangles When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle.Holt McDougal Geometry
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5-2 Bisectors of Triangles The circumcenter can be inside the triangle, outside the triangle, or on the triangle.Holt McDougal Geometry
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5-2 Bisectors of Triangles The circumcenter of ΔABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon.Holt McDougal Geometry
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5-2 Bisectors of Triangles Example 1: Using Properties of Perpendicular Bisectors DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC. G is the circumcenter of ∆ABC. By the Circumcenter Theorem, G is equidistant from the vertices of ∆ABC. GC = CB Circumcenter Thm. GC = 13.4 Substitute 13.4 for GB.Holt McDougal Geometry
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5-2 Bisectors of Triangles A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle .Holt McDougal Geometry
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5-2 Bisectors of Triangles Remember!The distance between a point and aline is the length of the perpendicularsegment from the point to the line.Holt McDougal Geometry
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5-2 Bisectors of Triangles Unlike the circumcenter, the incenter is always inside the triangle.Holt McDougal Geometry
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5-2 Bisectors of Triangles The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.Holt McDougal Geometry
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5-2 Bisectors of Triangles Example 3A: Using Properties of Angle Bisectors MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN. P is the incenter of ∆LMN. By the Incenter Theorem, P is equidistant from the sides of ∆LMN. The distance from P to LM is 5. So the distance from P to MN is also 5.Holt McDougal Geometry
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5-2 Bisectors of Triangles Example 3B: Using Properties of Angle BisectorsMP and LP are angle bisectorsof ∆LMN. Find m∠PMN. m∠MLN = 2m∠PLN PL is the bisector of ∠MLN. m∠MLN = 2(50°) = 100° Substitute 50° for m∠PLN. m∠MLN + m∠LNM + m∠LMN = 180° Δ Sum Thm. 100 + 20 + m∠LMN = 180 Substitute the given values. m∠LMN = 60° Subtract 120° from both sides. PM is the bisector of ∠LMN. Substitute 60° for m∠LMN.Holt McDougal Geometry
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5-2 Bisectors of Triangles CWHolt McDougal Geometry
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5-2 Bisectors of Triangles Objectives Apply properties of medians of a triangle. Apply properties of altitudes of a triangle.Holt McDougal Geometry
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5-2 Bisectors of Triangles Vocabulary median of a triangle centroid of a triangle altitude of a triangle orthocenter of a triangleHolt McDougal Geometry
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5-2 Bisectors of Triangles A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Every triangle has three medians, and the medians are concurrent.Holt McDougal Geometry
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5-2 Bisectors of Triangles The point of concurrency of the medians of a triangle is the centroid of the triangle . The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance.Holt McDougal Geometry
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5-2 Bisectors of Triangles An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle.Holt McDougal Geometry
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5-2 Bisectors of Triangles In ΔQRS, altitude QY is inside the triangle, but RX and SZ are not. Notice that the lines containing the altitudes are concurrent at P. This point of concurrency is the orthocenter of the triangle.Holt McDougal Geometry
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5-2 Bisectors of Triangles Helpful Hint The height of a triangle is the length of an altitude.Holt McDougal Geometry
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5-2 Bisectors of Triangles vocabulary The midsegment of a triangle - Segment that joins the midpoints of any two sides of a triangle.Holt McDougal Geometry
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5-2 Bisectors of Triangles Theorem The midsegment of a triangle is half the length of, and parallel to, the third side of a triangle.Holt McDougal Geometry
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