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.

7 _____ _____ _____ _____ 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. _____ _____ _____ _____ _____ is a _____. Step 2: Calculate the Distances of both diagonals to show they are equal. _____ _____ _____ is a _____.

8 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. 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.

9 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. 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).

10 Next, show that the legs of the trapezoid are congruent. 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.