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