5.3 - Concurrent Lines, Medians, and Altitudes. When three or more lines intersect in one point,...

5.3 - Concurrent Lines, Medians, and Altitudes

When three or more lines intersect in one point, they are concurrent. The point at which they intersect is the point of concurrency.

Point of concurrency

For any triangle, there are four different sets of lines of concurrency.

Theorem 5-6

The perpendicular bisector of the sides of a triangle are concurrent at a point equidistant from the vertices.

The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle.

A circle can be circumscribed about the triangle.

circumcenter

Theorem 5-7

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle.

incenter

Using the incenter, a circle can be inscribed in a triangle.

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

median

In a triangle, the point of concurrency of the medians is the centroid. The centroid is the center of gravity for a triangle – where it will balance.

centroid

Theorem 5-8 The medians of a triangle are concurrent at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.

DC = 2/3 DJ EC = 2/3 EG FC = 2/3 FH

F

G

D

H

EJ

C

An altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. An altitude can be a side of a triangle or it may lie outside the triangle.

Acute - altitude is inside.

Right - altitude is a side.

Obtuse - altitude is outside.

The altitudes are concurrent at the orthocenter of the triangle.

Orthocenter

Theorem 5-9 The lines that contain the altitudes of a triangle are concurrent.

SummaryPart of Triangle Point of

ConcurrencyResult

PerpendicularBisectors

Circumcenter Circumscribe acircle about thetriangle

Angle Bisectors Incenter Inscribe a circleinside the triangle

Medians Centroid Center of gravity

Altitudes Orthocenter Can be inside oroutside of triangle

Is AB a perpendicular bisector, an altitude, a median, an angle bisector, or none of these?

Is AB a perpendicular bisector, an altitude, a median, an angle bisector, or none of these?

Find the center of the circle that circumscribes the triangle. (Find the intersection of the perpendicular bisectors).

Find the center of the circle that circumscribes the triangle. (Find the intersection of the perpendicular bisectors).

A(-4, 5) B(-2, 5) C(-2, -2)

Homework

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Day 1: 8 – 22, 27 - 29Day 2: 1 - 7