Unit 5

Bisectors, Medians, and Altitudes Part 1

Vocab Concurrent Lines Point of Concurrency When three or more lines intersect they are called concurrent lines Point of Concurrency The point of intersection for concurrent lines

Perpendicular Bisector A line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side. Any point on a perpendicular bisector is equidistant from the endpoints of the segment.(theorem 5.1) Any point that is equidistant from the endpoints of a segment lies on the perpendicular bisector of that segment. (theorem 5.2) A triangle has three perpendicular bisectors.

Circumcenter The intersection of the three perpendicular bisectors, their point of concurrency. The circumcenter of a triangle is equidistant from the vertices of a triangle. (Theorem 5.3)

Angle Bisector Theorems Any point on the angle bisector is equidistant from the sides of the angle. (theorem 5.4) Any Point equidistant from the sides of an angle lies on the angle bisector. (theorem 5.5)

Incenter The intersection of the three angle bisectors. The incenter of a triangle is equidistant from each side of the triangle.

Median A segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.

Centroid The point of intersection of the three medians of a triangle The centroid is the point of balance for any triangle The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. (Theorem 5.7)

Centroid Example

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Altitude A segment formed from the vertex to the line containing the opposite side and is perpendicular to the opposite side. Every triangle has three altitudes, their intersection is know as the orthocenter.

Triangle Inequalities Part 2

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.(theorem 5.8)

Triangle Sides and Angles 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.

Example

Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Always check using the two smallest sides, they must be larger than the third. If this is true the numbers will represent a triangle.

Example Do these numbers represent a triangle? 1.) 9, 7, 12 Yes 2.) 5, 5, 10 No 3.) 1, 4, 6 4.) 6, 6, 2