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.

yx

mDCA = 155

mABC = 72

B

AC

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 =

2,

22121 yyxx

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?