5.1 Circumcenter and orthocenter
Theorem: A point lies on a perpendicular bisector of a segment if and only if it is equidistant from its _____
endpoints
Theorem: a point lies on the bisector of an angle if and only if it's equidistant from its ____
sides
For any real numbers a, b, and c, if a = b + c and c > 0, then a is greater than b (a>b)
Name the property
Comparison property
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Name the theorem
Triangle inequality theorem
The perpendicular bisectors of the sides of a triangle are concurrent at the circumcenter, the point that is equidistant from its vertices.
Name the theorem
Circumcenter theorem
A triangle's angle bisectors are concurrent at the incenter, the point that is equidistant from each side of the triangle.
Name the theorem
Incenter theorem
The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
Name the theorem
Exterior angle inequality theorem
If two sides of one triangle are congruent to two sides of a second triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.
Name the theorem
Hinge theorem
Define altitude
Altitude of a triangle is a segment that extends from a vertex and is perpendicular to the line containing the opposite side.
Define median
A segment extending from a vertex to the midpoint of the opposite side
If two sides of a triangle are not congruent, the larger angle is opposite the longer side.
Name the theorem
Greater angle theorem
If two sides of one triangle are congruent to two sides of a second triangle, and the third side of the first is longer than the third side of the second, then the angle opposite the third side of the first triangle is larger than the angle opposite the third side of the second triangle.
Name the theorem. (associates with the previous question)
Converse of the hinge theorem
The lines that contain the altitudes of a triangle are concurrent at the orthocenter.
Name the theorem
Orthocenter Theorem
The three medians of a triangle are concurrent and the centroid, which is two thirds of the distance from each vertex to the opposite side's midpoint.
Name the theorem
Centroid theorem
If two angles of a triangle are not congruent, the longer side is opposite the larger angle.
Name the theorem
Longer side theorem
An altitude may be where on a triangle?
Inside, outside, and on a side of the triangle
Define centroid
the intersection of a triangle's medians
The _______- is the longest side in a right triangle
Hypotenuse