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 CD A B

Theorem 5.2

Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

Example CD A 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 5.3 Circumcenter Theorem

The circumcenter of a triangle is equidistant from the vertices of the triangle.

Example C A B circumcenter K

Theorem 5.4

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

Theorem 5.5

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

Incenter

The point of concurrency of the angle bisectors.

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

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.

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 5.7 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.

Example C A B DL E F centroid

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 C A B D L E F orthocenter

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 a2a 3b + 2

The vertices of  QRS are Q(4, 6), R(7, 2), and S(1, 2). Find the coordinates of the orthocenter of  QRS.

Assignment: Pgs all