Relationships within Triangles Chapter 5

5.1 Midsegment Theorem and Coordinate Proof Midsegment of a Triangle- a segment that connects the midpoints of 2 sides of a triangle.

Midsegment Theorem The segment connecting the midpoints of 2 sides of a triangle is parallel to the 3 rd side and is half as long as that side.

Coordinate Proof Place geometric figures in a coordinate plane Use variables to represent the coordinates of the figure

Coordinate Proof of Midsegment Theorem

Example Graph ∆OHJ. O(0,0) H(m,n) J(m,0) Is ∆OHJ a right triangle? Find the lengths of the sides and the coordinates of each midpoint.

5.2 Use Perpendicular Bisectors Perpendicular Bisector Theorem Perpendicular Bisector Converse

Example Find x.

Example Three forest ranger stations are the same distance from the main office. How would you find the location of the main office?

5.3 Use Angle Bisectors Angle Bisector Theorem Angle Bisector Converse

Example Find x so that P lies on the bisector of <A.

Example G is the incenter of ∆RST. Find GW.

Proof of Angle Bisector Theorem

5.4 Use Medians and Altitudes Concurrency of Medians of a Triangle

Point of Concurrency 3 Special Segments Intersect inside, outside, or on the triangle Additional information Circumcenter Incenter centroid Orthocenter

5.5 Inequalities in One Triangle

Examples

Two sides of a triangle are 8 and 12 in length. Find the possible lengths for the 3 rd side.

5.6 Inequalities in Two Triangles Hinge Theorem Converse of Hinge Theorem

5.6 Indirect Proofs (Proof by Contradiction) Identify the statement you want to prove. Assume temporarily that this statement is false by assuming that its opposite is true. Reason logically until you reach a contradiction. Point out that the desired conclusion must be true because the contradiction proves the temporary assumption is false.

Example (Number Theory) Write an indirect proof that an odd number is not divisible by 4. Given: x is an odd number Prove: x is not divisible by 4

Example (Geometric) If an angle is an exterior angle of a triangle, prove that its measure is greater than the measure of either of its corresponding remote interior angles.

Example (Algebraic) If -3x + 4 > 16, then x < -4 Given: Prove: