5 1 Midsegment Theorem and Coordinate Proof Objectives

5. 1 Midsegment Theorem and Coordinate Proof Objectives: 1. To discover and use the Midsegment Theorem 2. To write a coordinate proof

Midsegment A midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments.

Example 1 Graph ΔACE with coordinates A( -1, -1), C(3, 5), and E(7, -5). Graph the midsegment MS that connects the midpoints of AC and CE.

Example 1 Now find the slope and length of MS and AE. What do you notice about the midsegment and the third side of the triangle?

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

Example 2 The diagram shows an illustration of a roof truss, where UV and VW are midsegments of ΔRST. Find UV and RS.

Example 3 1. 2.

Coordinate Proof Coordinate proofs are easy. You just have to conveniently place your geometric figure in the coordinate plane and use variables to represent each vertex. – These variables, of course, can represent any and all cases. – When the shape is in the coordinate plane, it’s just a simple matter of using formulas for distance, slope, midpoints, etc.

Example 4 Place a rectangle in the coordinate plane in such a way that it is convenient for finding side lengths. Assign variables for the coordinates of each vertex.

Example 4 Convenient placement usually involves using the origin as a vertex and lining up one or more sides of the shape on the x - or y-axis.

Example 5 Place a triangle in the coordinate plane in such a way that it is convenient for finding side lengths. Assign variables for the coordinates of each vertex.

Example 6 Place the figure in the coordinate plane in a convenient way. Assign coordinates to each vertex. 1. Right triangle: leg lengths are 5 units and 3 units 2. Isosceles Right triangle: leg length is 10 units

Example 7 A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex.

Example 8 Find the missing coordinates. The show that the statement is true.

Example 9 Write a coordinate proof for the Midsegment Theorem. Given: MS is a midsegment of ΔOWL Prove: MS || OL and MS = ½OL

Example 10 Explain why the choice of variables below might be slightly more convenient. Given: MS is a midsegment of ΔOWL Prove: MS || OL and MS = ½OL

Perpendicular Bisector A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector

Equidistant A point is equidistant from two figures if the point is the same distance from each figure. Examples: midpoints and parallel lines

5. 2: Special Segments Objectives: 1. To use and define perpendicular bisectors, angle bisectors, 2. To discover, use, and prove various theorems about perpendicular bisectors and angle bisectors

Perpendicular Bisector Theorem In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Converse of Perpendicular Bisector Theorem In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Example 1 Plan a proof for the Perpendicular Bisector Theorem.

Example 2 BD is the perpendicular bisector of AC. Find AD.

Example 3 Find the values of x and y.

Angle Bisector An angle bisector is a ray that 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 two sides of the angle.

Example 4 A soccer goalie’s position relative to the ball and goalposts forms congruent angles, as shown. Will the goalie have to move farther to block a shot toward the right goal post or the left one?

Example 5 Find the value of x.

Example 6 Find the measure of <GFJ. It’s not the Angle Bisector Theorem that could help us answer this question. It’s the converse. If it’s true.

Converse of the Angle Bisector Theorem If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

Example 7 For what value of x does P lie on the bisector of <A?

Warm-Up Three or more lines that intersect at the same point are called concurrent lines The point of intersection is called the point of concurrency

Example 1 Are the lines represented by the equations below concurrent? If so, find the point of concurrency. x+y=7 x + 2 y = 10 x-y=1

Points of Concurrency Objectives: 1. To define various points of concurrency 2. To discover, use, and prove various theorems about points of concurrency

Concurrency of Medians Theorem The medians of a triangle intersect at a point that is twothirds of the distance from each vertex to the midpoint of the opposite side.

Centroid The three medians of a triangle are concurrent. The point of concurrency is an interior point called the centroid It is the balancing point or center of gravity of the triangle.

Example 2 In ΔRST, Q is the centroid and SQ = 8. Find QW and SW.

Circumcenter Concurrency of Perpendicular Bisectors of a Triangle Theorem The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

Circumcenter The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter of the triangle. In each diagram, the circle circumscribes the triangle.

Incenter Concurrency of Angle Bisectors of a Triangle Theorem The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

Incenter The point of concurrency of the three angle bisectors of a triangle is called the incenter of the triangle. In the diagram, the circle is inscribed within the triangle.

Orthocenter Concurrency of Altitudes of a Triangle Theorem The lines containing the altitudes of a triangle are concurrent. G

Orthocenter The point of concurrency of all three altitudes of a triangle is called the orthocenter of the triangle. The orthocenter, P, can be inside, on, or outside of a triangle depending on whether it is acute, right, or obtuse, respectively.