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

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 xx x 35

And these…  Find the missing values.

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 =

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?