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Section 5-3: Concurrent Lines,
Medians, and Altitudes
March 6, 2012
Warm-up
Warm-up:
 Practice 5-1: 4-12
 Practice 5-2: 6-21
Warm-up
Practice 5-1: 4-12
Warm-up
Practice 5-1: 4-12
Warm-up
Practice 5-1: 4-12
Questions on Homework?
Section 5-3:
Concurrent Lines, Medians, and Altitudes
 Objectives: Today you will learn to
identify properties of
 perpendicular and angle bisectors, and
 medians and altitudes of a triangle.
Section 5-3: Bisectors of Triangles
Recall:
 Perpendicular Bisector: line or segment that is
perpendicular to a segment at its midpoint.
 Angle Bisector: ray, line or segment that
divides an angle into two congruent angles.
Section 5-3:
Median and Altitude of Triangles
 Median: segment whose endpoints are a
vertex and the midpoint
of the opposite side
 Altitude: perpendicular segment from a
vertex to the line containing
the opposite side
 Point of Concurrency: the
point at which three or more lines intersect
Section 5-3: Perpendicular Bisectors and
Acute Triangles
1. Label one paper “Perpendicular Bisector.”
2. Draw an acute triangle on the paper.
3. Fold the paper so that one side is exactly on top
of itself; so it is cut in half. That is the
perpendicular bisector.
4. Repeat with remaining two sides.
5. Where do the perpendicular bisectors intersect?
Section 5-3: Angle Bisectors and Acute
Triangles
1. Label the other paper “Angle Bisector.”
2. Draw an acute triangle on the paper.
3. Fold the paper so that one angle is cut in half.
That is the angle bisector.
4. Repeat with remaining two angles.
5. Where do the angle bisectors intersect?
Section 5-3:
Exploration with Geogebra
http://www.geogebra.org
Section 5-3: Theorems (p. 257-259)
•
Theorem 5-6: The perpendicular bisectors of the sides of a
triangle are concurrent at a point equidistant from the
vertices.
•
Theorem 5-7: The bisectors of the angles of a triangle are
concurrent at a point equidistant from the sides.
•
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.
•
Theorem 5-9: The lines that contain the altitudes of a
triangle are concurrent.
Section 5-3: Review
• Perpendicular Bisectors
• Angle Bisectors
• Medians
• Altitudes
Section 5-3: Hospital Location
Boise, ID; Helena, MT; and Salt Lake
City, UT, three large cities in the US,
want to build a new modern Hospital
that they can share.
But where should it be built?
Section 5-3: Hospital Location
Section 5-3: Hospital Location
Helena
(1, 6)
Salt Lake City
(2, -5)
Boise
(-4,0)
Section 5-3: Hospital Location
Pt of concurrency
(1.5, 0.5)
Wrap-up
 Today you learned to identify properties of
perpendicular and angle bisectors, and
medians and altitudes of a triangle.
 Tomorrow you’ll learn about Indirect
Reasoning
 Quiz on sections 5-1 to 5-3 on Thursday!
 Homework
p. 259-262: 1-5, 8-16, 19-22, 27-29, 33, 34