Download Medians, altitudes, and perpendicular bisectors

MEDIANS, ALTITUDES, AND
PERPENDICULAR BISECTORS
October 13, 2009
Objectives

Content
 Students
will learn about medians, altitudes,
perpendicular bisectors, and angle bisectors.

Language
 Students
will participate in discussion.
 Students will demonstrate understanding in writing.
Yesterday’s Homework
Yesterday’s Homework
Yesterday’s Homework
Yesterday’s Homework
Yesterday’s Homework
Yesterday’s Homework
Yesterday’s Homework
Yesterday’s Homework
Yesterday’s Homework
A little vocabulary review

Scalene triangle:________sides are congruent

Isosceles triangle:________sides are congruent

Equilateral triangle:________sides are congruent
More vocabulary review

Acute: ______ acute angle(s)

Obtuse: ______ obtuse angle(s)

Right: ______ right angle(s)

Equiangular: ______ angles are congruent
Find these measures.

Find the missing values.
x
x
x
x
35
And these…

Find the missing values.
B
mABC = 72
x
y
C
A
mDCA = 155
D
What does it mean to be congruent?


When two figures have the same shape and size,
they are called congruent.
What kinds of congruent triangles are there?
What are the parts of an isosceles
triangle?




Base
Legs
Vertex angle
Base angles
What is a median?

A median of a triangle is a segment from a vertex
to the midpoint of the opposite side.
What is an altitude?

An altitude of a triangle is the perpendicular
segment from a vertex to a line that contains the
opposite side.
More about altitudes



In a right triangle, two of the altitudes are part of the
triangle.
Since the two legs are perpendicular to each other
then each leg is also an altitude of the triangle.
The third altitude is inside the triangle.
Even more about altitudes


In an obtuse triangle, two of the altitudes are actually
outside the triangle.
This is why the definition states that the altitude is the
perpendicular segment from a vertex to the line that
contains the opposite side.
What is a perpendicular bisector?

A perpendicular bisector is a line (or ray or
segment) that is perpendicular to a segment at its
midpoint.
What is an angle bisector?

An angle bisector cuts an angle into two equal
parts.
What are concurrent line?

Concurrent lines are two or more lines that intersect
in one point.
Concurrent medians

The medians are concurrent at a point called the
centroid. The length of the segment from a vertex to
the centroid is always 2/3 the length of the entire
median.
Concurrent altitudes

The lines that contain the altitudes are
concurrent at a point called the orthocenter.
Concurrent perpendicular bisectors

The perpendicular bisectors of the sides are
concurrent at a point called the circumcenter.
Concurrent angle bisectors

Angle bisectors are rays running from the bisector
of each angle of the triangle. They all meet at the
incenter.
Working with these terms on a
coordinate plane

Midpoint formula =
 x1  x 2 y1  y 2 
,


2 
 2
Try this
Triangle RST has the
following coordinates:
R (-6, -8), S (6, 4), T (-6, 10)
 Find the midpoint of each
side.
 What else can we find?
