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Perpendicular Bisectors and Angle Bisectors
 Perpendicular Bisectors
A perpendicular bisector of a side of a triangle is a line, segment, or ray in the same plane as the triangle that is perpendicular to the side and passes through its midpoint.
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant, or the same distance, from the endpoints of the segment.
Converse of the Perpendicular Theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment
Two properties of perpendicular bisectors are:
- a point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment
- the three perpendicular bisectors of the sides of a triangle meet at a point, called the circumcenter of the triangle, that is equidistant from the three vertices of the triangle
The three perpendicular bisectors of the sides of a triangle meet in a single point. This point is the center of the circumscribed circle, which passes through each vertex of the triangle. Except for the three points where the circle touches the triangle, the circle is outside the triangle.
Circumcenter Theorem
The circumcenter of a triangle is equidistant from the vertices of the triangle.
Write the Equation of the Perpendicular Bisector of a Segment
Step 1: Find the midpoint of the segment
Step 2: Find the slope of the perpendicular bisector of the segment (opposite reciprocal of the slope of the original segment)
Step 3: Use the point-slope form to write the equation
If a triangle on a coordinate plane has two sides that lie along the axes, you can easily find the circumcenter. Find the equations for the perpendicular bisectors of those two sides. The intersection of their graphs is the circumcenter.
Find the coordinates of the circumcenter of a triangle by using...
Find the coordinates of the circumcenter of a triangle by using system of equations of the perpendicular bisectors.
Angle Bisectors
Another special segment, ray, or line is an angle bisector, which divides an angle into two congruent angles.
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.
Two properties of angle bisectors are:
- a point is on the angle bisector of an angle if and only if it is equidistant from the sides of the angle
- the three angle bisectors of a triangle meet at a point, called the incenter of the triangle, that is equidistant from the three sides of the triangle
The three angle bisectors of a triangle intersect in a single point called the incenter. This point is the center of a circle that just touches the three sides of the triangle. Except for the three points where the circle touches the sides, the circle is inside the triangle. The circle is said to be inscribed in the triangle.
Incenter Theorem
The incenter of a triangle is equidistant from the sides of the triangle.
Medians and Altitudes
Median
A median is a line segment that connects the vertex of a triangle to the midpoint of the opposite side. The three medians of a triangle intersect at the centroid of the triangle.
Centroid Theorem
The centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median
Find the lengths of the segments using the Centroid Theorem and...
Find the lengths of the segments using the Centroid Theorem and medians of a triangle
Altitude of a Triangle
An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. In other words, the height of a triangle is the length of an altitude. Every triangle has three altitudes. An altitude can be located inside, outside, or on the triangle.
The orthocenter of a triangle is the point where all three altitudes of the triangle meet.
Find the Orthocenter
Step 1: Graph the triangle
Step 2: Find the equations of the lines containing two altitudes of the triangle.
Step 3: Solve the system of equations of the two altitudes to find the coordinates of the orthocenter
Find the coordinates of the orthocenter of a triangle using system of...
Find the coordinates of the orthocenter of a triangle using system of equations of the altitudes of the triangle.
Inequalities and Triangles
Exterior Angle Inequality Theorem
If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles.
Angle - Side Relationships
When the sides of triangles are not congruent, there is a relationship between the sides and angles of the triangles.
- If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.
- If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. In other words, the sum of each pair of side lengths is greater than the third side length.
By contrast, to show that three lengths cannot be the side lengths of a triangle, you only need to show that one of the three triangle inequalities is false.
Distance Between a Point and a Line
The perpendicular segment from a point to a line is the shortest segment from the point to the line.
The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.
Inequalities Involving Two Triangles
SAS Inequality/Hinge Theorem
If two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle has a greater measure than the included angle in the other, then the third side of the first triangle is longer than the third side of the second triangle.
SSS Inequality
If two sides of a triangle are congruent to two sides of another triangle and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the other corresponding angle in the second triangle.
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Triangle Properties
