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.

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

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

MediansAltitudes

Homework

Work Packet: Concurrent Lines