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. 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.

2 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. 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.

3 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. 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.

4 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. 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. 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.

5 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. 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.

6 Show: Formula: Work Step 1: Calculate the Distances of all four sides to show that the opposite sides are equal. _____ _____ 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. _____ _____ _____ _____ Statement: _____ is a Rhombus because _____.

7 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. _____ _____ _____ _____ _____ 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.

8 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. 2. Prove that quadrilateral PLUS with the vertices P(2,1), L(6,3), U(5,5), and S(1,3) is a rectangle. 20 3. Prove that quadrilateral DAVE with the vertices D(2,1), A(6,-2), V(10,1), and E(6,4) is a rhombus. 4. Prove that quadrilateral GHIJ with the vertices G(-2,2), H(3,4), I(8,2), and J(3,0) is a rhombus. 21 5. Prove that a quadrilateral with vertices J(2,-1), K(-1,-4), L(-4,-1) and M(-1, 2) is a square. 6. Prove that ABCD is a square if A(1,3), B(2,0), C(5,1) and D(4,4). 22 Day 3 Using COORDINATE GEOMETRY to Prove Trapezoids Warm - Up 1.

9 2. 23 Proving a Quadrilateral is a Trapezoid Method: Show one pair of sides are parallel (same slope ) and one pair of sides are not parallel (different slopes). Example 1: Prove that KATE a trapezoid with coordinates K(0,4), A(3,6), T(6,2) and E(0,-2). Show: Formula: Work Calculate the Slopes of all four sides to show 2 sides are parallel and 2 sides are nonparallel. _____ _____ and _____ _____ Statement: _____ is a Trapezoid because _____. 24 Proving a Quadrilateral is an Isosceles Trapezoid Method: First, show one pair of sides are parallel (same slope ) and one pair of sides are not parallel (different slopes). Next, show that the legs of the trapezoid are congruent.

10 Example 2: Prove that quadrilateral MILK with the vertices M(1,3), I(-1,1), L(-1, -2), and K(4,3) is an isosceles trapezoid. Show: Formula: Work Step 1: Calculate the Slopes of all four sides to show Step 2: Calculate the distance of 2 sides are parallel and 2 sides are nonparallel. both non-parallel sides (legs) to show legs congruent. Statement: _____ is an Isosceles Trapezoid because _____. 22)()(yxd 25 Practice Prove that the quadrilateral with the vertices C(-3,-5), R(5,1), U(2,3) and D(-2,0) is a trapezoid but not an isosceles trapezoid. 26 Challenge SUMMARY Exit Ticket 27 Day 3 - Homework 1.