Section 5-3 Concurrent Lines, Medians, Altitudes SPI 32J: identify the appropriate segment of a triangle given a diagram and vs (median, altitude, angle and perpendicular bisector)

Objectives:• Identify properties of perpendicular and angle bisectors• Identify properties of medians and altitudes of triangles

Concurrent• three or more lines intersect in one point

Point of Concurrency• the point at which the concurrent lines intersect

Point of Concurrency

Concurrency and Perpendicular/Angle Bisectors

Theorem 5-6

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

Concurrency and Angle Bisectors

Theorem 5-7

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

Concurrency and Perpendicular Bisectors

The figure shows perpendicular bisectors concurrent at S.

The point S is called the circumcenter of the triangle.

Points A, B, and C are equidistant from point S. The circle is circumscribed about the triangle.

Concurrency and Angle Bisectors

The figure shows angle bisectors concurrent at I.

The point I is called the incenter of the triangle.

Points A, B, and C are equidistant from point I. The circle is inscribed in the triangle.

Find the center of the circle that circumscribes ∆XYZ.

Apply Perpendicular Bisectors

Find the perpendicular bisectors

(Line XY) y = 4

(Line XZ) y = 3

The lines y = 4 and x = 3 intersect at the point (3, 4).

This point is the center of the circle that circumscribes ∆XYZ.

City planners want to locate a fountain equidistant

from three straight roads that enclose a park.

Explain how they can find the location.

Theorem 5-7 states that the bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

The city planners should find the point of concurrency of the angle bisectors of the triangle formed by the three roads and locate the fountain there.

The roads form a triangle around the park.

Real-world and Angle Bisectors

Median of a Triangle

The point of concurrency of the medians is called centroid.Point G is the 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.

AG = 2/3 ADCG = 2/3 CFBG = 2/3 BE

Medians

The centroid is the point of concurrency of the medians of a triangle.

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. (Theorem 5-8)

WM = WX Theorem 5-823

16 = WX Substitute 16 for WM.23

24 = WX Multiply each side by .32

M is the centroid of ∆WOR, and WM = 16. Find WX.

Because M is the centroid of WOR, WM = WX.23

Apply Median of a Triangle

Altitude of a Triangle

In a triangle, the perpendicular from a vertex to the opposite side is called the Altitude.

Theorem 5-9

The lines that contain the altitudes of a triangle are concurrent.

The altitude can be a side of a triangle or may lie outside the triangle.

Altitude of a Triangle

Theorem 5-9

The lines that contain the altitudes of a triangle are concurrent.

The point where the altitudes are concurrent are called the orthocenter of the triangle.

Because LX = XM, point X is the midpoint of LM, and KX is a median of KLM.

Because KX is perpendicular to LM at point X, KX is an altitude.

So KX is both a median and an altitude.

Is KX a median, an altitude, neither, or both?

Altitude of a Triangle

Compare Medians and Altitudes

Median goes from vertex to midpoint of segment opposite.

Altitude is a perpendicular segment from vertex to

segment opposite.