Medians, altitudes, and perpendicular bisectors

May 1, 2008

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 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.