Concurrent Lines Geometry Mrs. King Unit 4, Day 7

Definition Concurrent – when three or more lines intersect in one point

Definition Point of Concurrency – the point at which lines are concurrent

On your triangle, complete a perpendicular bisector on all three sides.

Draw a Triangle What do you notice about the perpendicular bisectors of a triangle? What about the relationship between the point of concurrency and the vertices?

More Theorems Theorem 4-16: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

Theorem 4-16 Continued: Q The point of concurrency (point C) is called the circumcenter. The circumcenter is equidistant from all the vertices The circle is circumscribed around the triangle S R

On your triangle, complete a angle bisector on all three angles.

Draw a Triangle What do you notice about the angle bisectors of a triangle? What about the relationship between the point of concurrency and the sides?

More Theorems Theorem 4-17: The bisector of the angles of a triangle are concurrent at a point equidistant from the sides

Theorem 4-17 Continued: The point of concurrency (point I) is called the incenter. The incenter is equidistant from all the sides The circle is inscribed in the triangle U T V I

Definition Altitude of a triangle – a perpendicular segment from a vertex to the line containing the side opposite the vertex.

Altitudes An altitude of a triangle may lie outside the triangle:

More Theorems Theorem 4-18: The lines that contain the altitudes of a triangle are concurrent.

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

Are Medians Concurrent? Draw a triangle and then draw all three medians. Are they concurrent?

More Theorems Theorem 4-19: The medians of a triangle are concurrent.

Summary Perpendicular Bisectors Angle Bisectors Medians Altitudes

Homework Work Packet: Concurrent Lines