GEOMETRY COORDINATE GEOMETRY Proofs Name Period 1
Coordinate Proof Help Page Formulas Slope: Distance: To show segments are congruent: Use the distance formula to find the length of the sides and show that the sides have the same lengths and are therefore congruent. To show that lines are perpendicular: Find the slopes of the lines and show that their products is -1. To show that two lines are parallel: Find the slopes of the lines and show that they are equal. Right Triangle Method 1: Show that the Pythagorean Theorem works. Method : Show that two of the sides are perpendicular. Isosceles Triangle Show that sides are congruent. Equilateral Triangle Show that all three sides are congruent. Rhombus Show that all sides are congruent. Parallelogram Method 1: Show that both pairs opposite sides are parallel. Method : Show that both pairs of opposite sides are congruent. Rectangle Method 1: Show that both pairs of opposite sides are parallel and there is a right angle. Method : Show that both pairs of opposite sides are congruent and diagonals are congruent. Square Method 1: Show that all four sides are congruent and there is a right angle. Method : Show that all four sides are congruent and diagonals are congruent. Trapezoid Show that one pair of sides are parallel and the other are not. Isosceles Trapezoid Show that the polygon is a trapezoid (see above) and show that the nonparallel sides are congruent.
Table of Contents Day 1 Essential Question: How can I use coordinate geometry to prove right triangles and parallelograms In class: Pages 5-7 Homework: Page 8 Day Essential Question: How can I use coordinate geometry to prove rectangles, rhombi, and squares. In class: Pages 9-11 Homework: Page 1 Day 3 Essential Question: How can I use coordinate geometry to prove trapezoids. In class: Pages 13-14 Homework: Page 15 Day 4: Students will practice writing coordinate geometry proofs (review) In class: Pages 16-18 Homework: Finish anything that has not yet been completed. 3
Warm-Ups Day 1: Graph AB with coordinates A(-, -5) and B(6,-1). Graph CD with coordinates C(-1,3) and D(,-3). Find the slope of AB and CD Are the lines parallel or perpendicular? Day : 1. QUVX is a rectangle. Find the value of x and y.. Find the value of a, b, and c in the rhombus. Day 3: 4
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, ) is a right triangle. Show: Formula: d ( x) ( y ) Work: Calculate the Distances of all three sides to show Pythagorean s Theorem is true. a + b = c ( ) + ( ) = ( ) Statement: 5
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.. Prove that A (-, -), B (5, -1), C (1, ) is a an isosceles right triangle. Show: Formula: d ( x) ( y ) Work: Calculate the Distances of all three sides to show Pythagorean s Theorem is true. a + b = c ( ) + ( ) = ( ) Statement: 6
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(-,3), M(4,3), N(,-) and O(-4,-) is a parallelogram. Show: Formula: d ( x) ( y ) Work: Calculate the Distances of all four sides to show that the opposite sides are equal. AND Statement: is a parallelogram because. 7
Homework- Day 1 1.. 3. 8
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: d ( x) ( y ) Work Step 1: Calculate the Distances of all four sides to show that the opposite sides are equal. AND is a parallelogram because. Step : Calculate the Distances of both diagonals to show they are equal. Statement: is a rectangle because. 9
Proving a Quadrilateral is a Rhombus Method: Prove that all four sides are congruent. Examples:. 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: d ( x) ( y ) Work Step 1: Calculate the Distances of all four sides to show that all four sides equal. Statement: is a Rhombus because. 10
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(,-4) is a square. Show: Formula: d ( x) ( y ) Work Step 1: Calculate the Distances of all four sides to show all sides are equal. is a. Step : Calculate the Distances of both diagonals to show they are equal. is a. Statement: is a Square because. 11
Homework- Day 1. Prove that quadrilateral ABCD with the vertices A(,1), B(1,3), C(-5,0), and D(-4,-) is a rectangle.. Prove that quadrilateral DAVE with the vertices D(,1), A(6,- ), V(10,1), and E(6,4) is a rhombus. 3. Prove that ABCD is a square if A(1,3), B(,0), C(5,1) and D(4,4). 1
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,) and E(0,-). Show: Formula: Work Calculate the Slopes of all four sides to show sides are parallel and sides are nonparallel. and Statement: is a Trapezoid because. 13
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. Example : Prove that quadrilateral MILK with the vertices M(1,3), I(-1,1), L(-1, -), and K(4,3) is an isosceles trapezoid. Show: Formula: d ( x) ( y ) Work Step 1: Calculate the Slopes of all four sides to show Step : Calculate the distance of sides are parallel and sides are nonparallel. both non-parallel sides (legs) to show legs congruent. Statement: is an Isosceles Trapezoid because. 14
Homework- Day 3 1.. 15
Day 4 Practice writing Coordinate Geometry Proofs 1. The vertices of ABC are A(3,-3), B(5,3) and C(1,1). Prove by coordinate geometry that ABC is an isosceles right triangle.. Given ABC with vertices A(-4,), B(4,4) and C(,-6), the midpoints of AB and BC are P and Q, respectively, and PQ is drawn. Prove by coordinate geometry: a. PQ AC b. PQ = ½ AC 16
3. Quadrilateral ABCD has vertices A(-6,3), B(-3,6), C(9,6) and D(-5,-8). Prove that quadrilateral ABCD is: c. a trapezoid d. not an isosceles trapezoid 4. The vertices of quadrilateral ABCD are A(-3,-1), B(6,), C(5,5) and D(-4,). Prove that quadrilateral ABCD is a rectangle. 17
5. The vertices of quadrilateral ABCD are A(-3,1), B(1,4), C(4,0) and D(0,-3). Prove that quadrilateral ABCD is a square. 6. Quadrilateral METS has vertices M(-5, -), E(-5,3), T(4,6) and S(7,). Prove by coordinate geometry that quadrilateral METS is an isosceles trapezoid. 18