Transcript Concurrent Lines, Medians, and Altitudes LESSON 5-3 Additional Examples Find the center of the circle that circumscribes XYZ. Because X has coordinates (1, 1) and.

Concurrent Lines, Medians, and Altitudes
LESSON 5-3
Additional Examples
Find the center of the circle that circumscribes
XYZ.
Because X has coordinates (1, 1) and Y has
coordinates (1, 7), XY lies on the vertical line x = 1.
The perpendicular bisector of XY is the horizontal line
that passes through (1, 1 + 7 ) or (1, 4), so the equation
2
of the perpendicular bisector of XY is y = 4.
Because X has coordinates (1, 1) and Z has coordinates (5, 1), XZ lies on
the horizontal line y = 1. The perpendicular bisector of XZ is the vertical line
that passes through ( 1 + 5 , 1) or (3, 1), so the equation of the perpendicular
2
bisector of XZ is x = 3.
HELP
GEOMETRY
Concurrent Lines, Medians, and Altitudes
LESSON 5-3
Additional Examples
(continued)
The lines y = 4 and x = 3 intersect at the point (3, 4).
This point is the center of the circle that circumscribes
XYZ.
Quick Check
HELP
GEOMETRY
Concurrent Lines, Medians, and Altitudes
LESSON 5-3
Additional Examples
City planners want to locate a fountain equidistant from three
straight roads that enclose a park. Explain how they can find the
location.
The roads form a triangle around the park.
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.
Quick Check
HELP
GEOMETRY
Concurrent Lines, Medians, and Altitudes
LESSON 5-3
Additional Examples
M is the centroid of
WOR, and WM = 16. Find WX.
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)
Because M is the centroid of
2
WM = 3 WX
16 =
2
WX
3
24 = WX
HELP
2
WOR, WM = 3 WX.
Theorem 5-8
Substitute 16 for WM.
3
Multiply each side by 2 .
Quick Check
GEOMETRY
Concurrent Lines, Medians, and Altitudes
LESSON 5-3
Additional Examples
Is KX a median, an altitude, neither, or both?
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.
HELP
Quick Check
GEOMETRY