Lesson 5 1 Bisectors Medians and Altitudes Ohio

Lesson 5 -1 Bisectors, Medians, and Altitudes

Ohio Content Standards:

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

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

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

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

Concurrent Lines When three or more lines intersect at a common point.

Point of Concurrency

Point of Concurrency The point of intersection where three or more lines meet.

Circumcenter

Circumcenter The point of concurrency of the perpendicular bisectors of a triangle.

Theorem 5. 3 Circumcenter Theorem

Theorem 5. 3 Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle.

Example B A circumcenter K C

Theorem 5. 4

Theorem 5. 4 Any point on the angle bisector is equidistant from the sides of the angle. B A C

Theorem 5. 5

Theorem 5. 5 Any point equidistant from the sides of an angle lies on the angle bisector. B A C

Incenter

Incenter The point of concurrency of the angle bisectors.

Theorem 5. 6 Incenter Theorem

Theorem 5. 6 Incenter Theorem The incenter of a triangle is equidistant from each side of the triangle. incenter B P Q K A R C

Theorem 5. 6 Incenter Theorem The incenter of a triangle is equidistant from each side of the triangle. incenter B P Q K A R If K is the incenter of ABC, then KP = KQ = KR. C

Median

Median A segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.

Centroid

Centroid The point of concurrency for the medians of a triangle.

Theorem 5. 7 Centroid Theorem

Theorem 5. 7 Centroid 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.