5 8 Vocabulary coordinate proof Coordinate proofs use

5. 8 Vocabulary coordinate proof Coordinate proofs use coordinate geometry to prove geometric relationships or properties for figures. General Case: proves properties/relationships of figures which are specified in terms of variables: any square, any rectangle, any rhombus, etc. Properties will always be true Specific Case: proves properties/relationships for figures given by actual coordinates. Properties are only certain for that case.

General: To prove the general case you assign variables as the coordinates of the figure. If a coordinate proof requires calculations with fractions, choose coordinates that make the calculations simpler. For example, use multiples of two when you are to find coordinates of a midpoint (2 a, 2 b, etc. )

Positioning Shapes

Once the figure is placed in the coordinate plane, you can use Slope, Distance, or the Midpoint Formulas to prove statements about the figure. Slope: m= Mid. Point: M = Distance: d =

Example 1 A: Assigning Coordinates to Vertices Position each figure in the coordinate plane and give the coordinates of each vertex. rectangle with length twice the width

Example 1 B: Assigning Coordinates to Vertices Position each figure in the coordinate plane and give the coordinates of each vertex. A right triangle with legs of lengths s and t Position a square with side length 4 p in the coordinate plane and give the coordinates of each vertex.

Example 2: Specific Figures in the Coordinate Plane Position a square with a side length of 6 units in the coordinate plane. You can put one corner of the square at the origin. Position a right triangle with leg lengths of 2 and 4 units in the coordinate plane. (Hint: Use the origin as the vertex of the right angle. )

Example 3: Specific Coordinate Proof Given: D(– 4, 4), E(– 2, 1), F(– 6, 1), G(3, 1), H(5, – 2), J(1, – 2). Prove: ∆DEF ∆GHJ

Example 4: Using CPCTC In the Coordinate Plane Given: D(– 5, – 5), E(– 3, – 1), F(– 2, – 3), G(– 2, 1), H(0, 5), and I(1, 3) Prove: DEF GHI

Example 4: General Coordinate Proof Given: Rectangle PQRS Prove: The diagonals are

Lesson Quiz: Part I Position each figure in the coordinate plane. 1. rectangle with a length of 6 units and a width of 3 units 2. square with side lengths of 5 a units

Lesson Quiz: Part II 3. Given: Rectangle ABCD A(0, 0), B(0, 8), C(5, 8), and D(5, 0). E is mdpt. of BC, F is mdpt. of AD. Prove: EF = AB 4. Honors: Show that if a segment is drawn from the vertex of the right angle to the midpoint of the hypotenuse it divides the triangle into two triangles which each have an area equal to half the area of the right triangle.