GEOMETRY COORDINATE GEOMETRY Proofs

1 GEOMETRY COORDINATE GEOMETRY Proofs Name _____ Period _____ Table of Contents Day 1: SWBAT: Use COORDINATE GEOMETRY to Prove Right Triangles and Parallelograms Pgs: 2 8 HW: Pgs: 9 12 Day 2: SWBAT: Use COORDINATE GEOMETRY to Prove Rectangles, Rhombi, and Squares Pgs: 13 - 18 HW: Pgs: 19 21 Day 3: SWBAT: Use COORDINATE GEOMETRY to Prove Trapezoids Pgs: 22 - 26 HW: Pgs: 27 28 Day 4: SWBAT: Practice Writing COORDINATE GEOMETRY Proofs (REVIEW) Pgs: 29 - 31 Day 5: TEST 1 COORDINATE GEOMETRY Proofs Slope: We use slope to show parallel lines and perpendicular lines.

2 Parallel Lines have the same slope Perpendicular Lines have slopes that are negative reciprocals of each other. Midpoint: We use midpoint to show that lines bisect each other. Lines With the same midpoint bisect each other Midpoint Formula: 2121,22yyxxmid Distance: We use distance to show line segments are equal. You can use the Pythagorean Theorem or the formula: 22)()(yxd 2 Day 1 Using COORDINATE GEOMETRY To Prove Right Triangles and Parallelograms Warm Up Explaination:_____ Explaination:_____3 Proving a triangle is a right triangle Method: Calculate the distances of all three sides and then test the Pythagorean s theorem to show the three lengths make the Pythagorean s theorem true.

3 How to Prove Right Triangles 1. Prove that A (0, 1), B (3, 4), C (5, 2) is a right triangle. Show: Formula: Work: Calculate the Distances of all three sides to show Pythagorean s Theorem is true. a2 + b2 = c2 ( )2 + ( )2 = ( )2 Statement: 22)()(yxd 4 How to Prove an Isosceles Right Triangles Method: Calculate the distances of all three sides first, next show two of the three sides are congruent, and then test the Pythagorean s theorem to show the three lengths make the Pythagorean s theorem true.

4 2. Prove that A (-2, -2), B (5, -1), C (1, 2) is a an isosceles right triangle. Show: Formula: Work: Calculate the Distances of all three sides to show Pythagorean s Theorem is true. _____ _____ a2 + b2 = c2 ( )2 + ( )2 = ( )2 Statement: 22)()(yxd 5 Proving a Quadrilateral is a Parallelogram Method: Show both pairs of opposite sides are equal by calculating the distances of all four sides. Examples 3. Prove that the quadrilateral with the coordinates L(-2,3), M(4,3), N(2,-2) and O(-4,-2) is a parallelogram.

5 Show: Formula: Work: Calculate the Distances of all four sides to show that the opposite sides are equal. _____ _____ AND _____ _____ Statement: _____ is a parallelogram because _____. 22)()(yxd 6 PRACTICE SECTION: Example 1: Prove that the polygon with coordinates A(1, 1), B(4, 5), and C(4, 1) is a right triangle. Example 2: Prove that the polygon with coordinates A(4, -1), B(5, 6), and C(1, 3) is an isosceles right triangle.

6 7 Example 3: Prove that the quadrilateral with the coordinates P(1,1), Q(2,4), R(5,6) and S(4,3) is a parallelogram. Challenge 8 SUMMARY Proving Right Triangles Proving Parallelograms Exit Ticket 9 Homework 1. 2. 10 3. 4. 11 6. Prove that quadrilateral LEAP with the vertices L(-3,1), E(2,6), A(9,5) and P(4,0) is a parallelogram.

7 12 7. 8. 9. 13 Day 2 Using COORDINATE GEOMETRY to Prove Rectangles, Rhombi, and Squares Warm Up 1. 2. 14 Proving a Quadrilateral is a Rectangle Method: First, prove the quadrilateral is a parallelogram, then that the diagonals are congruent. Examples: 1. Prove a quadrilateral with vertices G(1,1), H(5,3), I(4,5) and J(0,3) is a rectangle. Show: Formula: Work Step 1: Calculate the Distances of all four sides to show that the opposite sides are equal.

8 _____ _____ AND _____ _____ _____ is a parallelogram because _____. Step 2: Calculate the Distances of both diagonals to show they are equal. _____ _____ Statement: _____ is a rectangle because _____. 22)()(yxd 15 Proving a Quadrilateral is a Rhombus Method: Prove that all four sides are congruent. Examples: 2. Prove that a quadrilateral with the vertices A(-1,3), B(3,6), C(8,6) and D(4,3) is a rhombus. Show: Formula: Work Step 1: Calculate the Distances of all four sides to show that all four sides equal.

9 _____ _____ _____ _____ Statement: _____ is a Rhombus because _____. 22)()(yxd 16 Proving that a Quadrilateral is a Square Method: First, prove the quadrilateral is a rhombus by showing all four sides is congruent; then prove the quadrilateral is a rectangle by showing the diagonals is congruent. Examples: 3. Prove that the quadrilateral with vertices A(-1,0), B(3,3), C(6,-1) and D(2,-4) is a square. Show: Formula: Work Step 1: Calculate the Distances of all four sides to show all sides are equal.

10 _____ _____ _____ _____ _____ is a _____. Step 2: Calculate the Distances of both diagonals to show they are equal. _____ _____ _____ is a _____. Statement: _____ is a Square because _____. 22)()(yxd 17 SUMMARY Proving Rectangles Proving Rhombi Proving Squares 18 Challenge Exit Ticket 1. 2. 19 DAY 2 - Homework 1. Prove that quadrilateral ABCD with the vertices A(2,1), B(1,3), C(-5,0), and D(-4,-2) is a rectangle.