Lesson 4-7 Triangles and Coordinate Proof Coordinate proof- uses figures in a coordinate plane and Algebra to prove geometric concepts. Placing Figures in the Coordinate Plane Use the origin as the vertex or the center of the figure Place at least one side of a polygon on an axis Keep the figure within the 1st quadrant, if possible Use coordinates that make math easy

Use the origin as vertex X of the triangle. Position and label right triangle XYZ with leg d units long on the coordinate plane. Use the origin as vertex X of the triangle. Place the base of the triangle along the positive x-axis. Position the triangle in the first quadrant. Since Z is on the x-axis, its y-coordinate is 0. Its x-coordinate is d because the base is d units long. X (0, 0) Z (d, 0) Example 7-1a

Since triangle XYZ is a right triangle the x-coordinate of Y is 0 Since triangle XYZ is a right triangle the x-coordinate of Y is 0. We cannot determine the y-coordinate so call it b. Answer: Y (0, b) X (0, 0) Z (d, 0) Example 7-1b

Position and label equilateral triangle ABC with side w units long on the coordinate plane. Answer: Example 7-1c

Name the missing coordinates of isosceles right triangle QRS. Q is on the origin, so its coordinates are (0, 0). The x-coordinate of S is the same as the x-coordinate for R, (c, ?). The y-coordinate for S is the distance from R to S. Since QRS is an isosceles right triangle, The distance from Q to R is c units. The distance from R to S must be the same. So, the coordinates of S are (c, c). Answer: Q(0, 0); S(c, c) Example 7-2a

Name the missing coordinates of isosceles right ABC. Answer: C(0, 0); A(0, d) Example 7-2b

Write a coordinate proof to prove that the segment that joins the vertex angle of an isosceles triangle to the midpoint of its base is perpendicular to the base. Example 7-3a

Given: XYZ is isosceles. The first step is to position and label a right triangle on the coordinate plane. Place the base of the isosceles triangle along the x-axis. Draw a line segment from the vertex of the triangle to its base. Label the origin and label the coordinates, using multiples of 2 since the Midpoint Formula takes half the sum of the coordinates. Given: XYZ is isosceles. Prove: Example 7-3b

Proof: By the Midpoint Formula, the coordinates of W, the midpoint of , is The slope of or undefined. The slope of is therefore, . Example 7-3c

Write a coordinate proof to prove that the segment drawn from the right angle to the midpoint of the hypotenuse of an isosceles right triangle is perpendicular to the hypotenuse. Example 7-3d

Proof: The coordinates of the midpoint D are The slope of is or 1. The slope of or –1, therefore . Example 7-3e

or undefined. The slope of or 0, therefore DEF is a right triangle. DRAFTING Write a coordinate proof to prove that the outside of this drafter’s tool is shaped like a right triangle. The length of one side is 10 inches and the length of another side is 5.75 inches. Proof: The slope of or undefined. The slope of or 0, therefore DEF is a right triangle. The drafter’s tool is shaped like a right triangle. Example 7-4a

FLAGS Write a coordinate proof to prove this flag is shaped like an isosceles triangle. The length is 16 inches and the height is 10 inches. C Example 7-4b

Determine the lengths of CA and CB. Proof: Vertex A is at the origin and B is at (0, 10). The x-coordinate of C is 16. The y-coordinate is halfway between 0 and 10 or 5. So, the coordinates of C are (16, 5). Determine the lengths of CA and CB. Since each leg is the same length, ABC is isosceles. The flag is shaped like an isosceles triangle. Example 7-4c