Transparency 2 Review: Lesson 4-6 Mini-Quiz

Class Greeting

Objective: The students will be able to write Coordinate Proofs.

Chapter 4 – Lesson 7 Triangles and Co-ordinate Proof

Vocabulary Coordinate Proof A proof of a geometric concept using an algebraic proof and the coordinate plane

Setting up for Coordinate Proof Use the origin as a vertex or center Place at least one side on an axis Stay in first quadrant if possible Try to make the coordinate calculations simple

Example 7-1a Use the origin as vertex X of the triangle. Place the base of the triangle along the positive x-axis. Position and label right triangle XYZ with leg d units long on the coordinate plane. X (0, 0) Z (d, 0) 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. Since triangle XYZ is a right triangle the x-coordinate of Y is 0. We cannot determine the y-coordinate so call it b. Y (0, b) Answer:

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

Example 7-2a Name the missing coordinates of isosceles right triangle QRS. Answer: Q(0, 0); S(c, c) 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).

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

Example 7-3a 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. The first step is to position and label an isosceles 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 Prove: Given:  XYZ is isosceles. (Continued) Proof: By the Midpoint Formula, the coordinates of W, the midpoint of, is The slope of or undefined. The slope of is therefore,.

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

Lesson Summary: Objective: The students will be able to write Coordinate Proofs.

Preview of the Next Lesson: Objective: The students will complete a Practice Test on Chapter 4.

Homework Geometry 4-7