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 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 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.