Relationships Within Triangles Chapter5

Triangle Midsegment Theorem If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length. A C B D E 30 cm 60 cm

In ∆XYZ, M, N, and P are the midpoints. The perimeter of ∆MNP is 60. Find NP, XY, XZ and YZ. X P Z N Y M NP = NP = NP = 14 24(2) = 48 22(2) = (2) = 28

Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the end points of the segment. Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. A B C D If then D is the midpoint of.

Angle Bisector Theorem If a point is on the bisector of an angle, then the point is equidistant to 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 the point is on the angle bisector. F G H J

Vocabulary Concurrent – when 3 or more lines intersect in one point. Point of concurrency Circumcenter of the triangle- The point of concurrency of the perpendicular bisectors of a triangle. X X is the circumcenter of the triangle.

Theorem 5-6 The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. QX = PX = RX X P QR X is the center of the circle, which is circumscribed about the triangle.

Theorem 5-7 The bisectors of the angles of a triangle are concurrent at a point that is equidistant from the sides. The point of concurrency is called the incenter. T Y V Z U X I XI = YI = ZI

Vocabulary Median – a segment whose endpoints are a vertex and the midpoint of the opposite side. Every triangle has 3 medians that intersect inside the triangle.

Altitude – the perpendicular segment from a vertex to the line containing the opposite side. There are 3 altitudes in every triangle. Unlike bisectors and medians, an altitude can be a side of the triangle or it may be outside of the triangle. Acute Triangle: Altitude is inside. Right Triangle: Altitude is a side. Obtuse Triangle: Altitude is outside.

Inequalities in Triangles If no sides or angles are congruent, the side opposite the largest angle is the longest side; the side opposite the smallest angle is the shortest side. The opposite is also true. The angle opposite the longest side is the largest angle and the angle opposite the shortest side is the smallest angle. A B C 90º 57º 33º D E F

Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third. X Y Z XY + XZ > YZ XY + YZ > XZ XZ + YZ > XY