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Lesson 5-1 Bisectors, Medians, and Altitudes
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Ohio Content Standards:
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Formally define geometric figures.
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Ohio Content Standards: Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles of polygons; b. segments related to triangles (median, altitude, midsegment); c. points of concurrency related to triangles (centroid, incenter, orthocenter, and circumcenter);
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Perpendicular Bisector
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A line, segment, or ray that passes through the midpoint of the side of a triangle and is perpendicular to that side.
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Theorem 5.1
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Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
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Example CD A B
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Theorem 5.2
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Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.
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Example CD A B
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Concurrent Lines
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When three or more lines intersect at a common point.
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Point of Concurrency
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The point of intersection where three or more lines meet.
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Circumcenter
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The point of concurrency of the perpendicular bisectors of a triangle.
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Theorem 5.3 Circumcenter Theorem
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The circumcenter of a triangle is equidistant from the vertices of the triangle.
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Example C A B circumcenter K
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Theorem 5.4
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Any point on the angle bisector is equidistant from the sides of the angle. A C B
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Theorem 5.5
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Any point equidistant from the sides of an angle lies on the angle bisector. A B C
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Incenter
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The point of concurrency of the angle bisectors.
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Theorem 5.6 Incenter Theorem
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The incenter of a triangle is equidistant from each side of the triangle. C A Bincenter K P Q R
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Theorem 5.6 Incenter Theorem The incenter of a triangle is equidistant from each side of the triangle. C A Bincenter K P Q R If K is the incenter of ABC, then KP = KQ = KR.
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Median
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A segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.
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Centroid
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The point of concurrency for the medians of a triangle.
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Theorem 5.7 Centroid Theorem
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The centroid of a triangle is located two-thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median.
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Example C A B DL E F centroid
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Altitude
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A segment from a vertex in a triangle to the line containing the opposite side and perpendicular to the line containing that side.
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Orthocenter
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The intersection point of the altitudes of a triangle.
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Example C A B D L E F orthocenter
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Points U, V, and W are the midpoints of YZ, ZX, and XY, respectively. Find a, b, and c. Y W U X V Z 7.4 5c 8.7 15.2 2a2a 3b + 2
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The vertices of QRS are Q(4, 6), R(7, 2), and S(1, 2). Find the coordinates of the orthocenter of QRS.
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Assignment: Pgs. 243-245 13-20 all
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