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GEOMETRYCOORDINATEGEOMETRYProofsNamePeriod

Table of ContentsDay 1: SWBAT: Use Coordinate Geometry to Prove Right Triangles and ParallelogramsPgs: 2 – 8HW: Pgs: 9 – 12Day 2: SWBAT: Use Coordinate Geometry to Prove Rectangles, Rhombi, and SquaresPgs: 13 - 18HW: Pgs: 19 – 21Day 3: SWBAT: Use Coordinate Geometry to Prove TrapezoidsPgs: 22 - 26HW: Pgs: 27 – 28Day 4: SWBAT: Practice Writing Coordinate Geometry Proofs (REVIEW)Pgs: 29 - 31Day 5: TEST

Coordinate Geometry ProofsSlope: We use slope to show parallel lines and perpendicular lines.Parallel Lines have the same slopePerpendicular 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 otherMidpoint Formula:mid x1 x2 , y1 y 2 2 2Distance: We use distance to show line segments are equal.You can use the Pythagorean Theorem or the formula:d ( x) 2 ( y) 21

Day 1 – Using Coordinate Geometry To Prove Right Triangles andParallelogramsWarm – UpExplaination:Explaination:2

Proving a triangle is a right triangleMethod: Calculate the distances of all three sides and then test the Pythagorean’s theorem toshow 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:d ( x) 2 ( y) 2Work: Calculate the Distances of all three sides to show Pythagorean’s Theorem is true.a 2 b2 c 2()2 ()2 ()2Statement:3

“How to Prove an Isosceles Right Triangles”Method: Calculate the distances of all three sides first, next show two of the three sides arecongruent, and then test the Pythagorean’s theorem to show the three lengths make thePythagorean’s theorem true.2. Prove that A (-2, -2), B (5, -1), C (1, 2) is aan isosceles right triangle.Show:Formula:d ( x) 2 ( y) 2Work: Calculate the Distances of all three sides to show Pythagorean’s Theorem is true. a 2 b2 c 2()2 ()2 ()2Statement:4

Proving a Quadrilateral is a ParallelogramMethod: Show both pairs of opposite sides are equal by calculating the distances of all four sides.Examples3. Prove that the quadrilateral with the coordinates L(-2,3), M(4,3), N(2,-2) and O(-4,-2) is aparallelogram.Show:Formula:d ( x) 2 ( y) 2Work: Calculate the Distances of all four sides to show that the opposite sides are equal. AND Statement: is a parallelogram because .5

PRACTICE SECTION:Example 1: Prove that the polygon with coordinates A(1, 1), B(4, 5), and C(4, 1) is a righttriangle.Example 2: Prove that the polygon with coordinates A(4, -1), B(5, 6), and C(1, 3) is anisosceles right triangle.6

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.Challenge7

SUMMARYProving Right TrianglesProving ParallelogramsExit Ticket8

Homework1.2.9

3.4.10

6. Prove that quadrilateral LEAP with the verticesL(-3,1), E(2,6), A(9,5) and P(4,0) is a parallelogram.11

7.8.9.12

Day 2 – Using Coordinate Geometry to Prove Rectangles, Rhombi, and SquaresWarm – Up1.2.13

Proving a Quadrilateral is a RectangleMethod: 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) 2 ( y) 2WorkStep 1: Calculate the Distances of all four sides to showthat 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 .14

Proving a Quadrilateral is a RhombusMethod: 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:d ( x) 2 ( y) 2WorkStep 1: Calculate the Distances of all four sides to showthat all four sides equal. Statement: is a Rhombus because .15

Proving that a Quadrilateral is a SquareMethod: First, prove the quadrilateral is a rhombus by showing all four sides is congruent; thenprove 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:d ( x) 2 ( y) 2WorkStep 1: Calculate the Distances of all four sides to showall 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 .16

SUMMARYProving RectanglesProving RhombiProving Squares17

ChallengeExit Ticket1.2.18

DAY 2 - Homework1. Prove that quadrilateral ABCD with the vertices A(2,1), B(1,3), C(-5,0), andD(-4,-2) is a rectangle.2. Prove that quadrilateral PLUS with the vertices P(2,1), L(6,3), U(5,5), andS(1,3) is a rectangle.19

3. Prove that quadrilateral DAVE with the vertices D(2,1), A(6,-2), V(10,1), and E(6,4) is arhombus.4. Prove that quadrilateral GHIJ with the vertices G(-2,2), H(3,4), I(8,2), andJ(3,0) is a rhombus.20

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).21

Day 3 – Using Coordinate Geometry to Prove TrapezoidsWarm - Up1.2.22

Proving a Quadrilateral is a TrapezoidMethod: Show one pair of sides are parallel (same slope) and one pair of sides are notparallel (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:WorkCalculate the Slopes of all four sides to show2 sides are parallel and 2 sides are nonparallel. andStatement: is a Trapezoid because .23

Proving a Quadrilateral is an Isosceles TrapezoidMethod: First, show one pair of sides are parallel (same slope) and one pair of sides arenot parallel (different slopes). 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), andK(4,3) is an isosceles trapezoid.Show:Formula:d ( x) 2 ( y) 2WorkStep 1: Calculate the Slopes of all four sides to show2 sides are parallel and 2 sides are nonparallel.Step 2: Calculate the distance ofboth non-parallel sides(legs) to show legs congruent.Statement: is an Isosceles Trapezoid because .24

PracticeProve that the quadrilateral with the vertices C(-3,-5), R(5,1), U(2,3) and D(-2,0) is a trapezoidbut not an isosceles trapezoid.25

ChallengeSUMMARYExit Ticket26

Day 3 - Homework1.2.27

3. .4. Triangle TOY has coordinates T(-4, 2), O(-2, -2), and Y(2, 0). Prove TOY is an isoscelesright triangle.28

Day 4 – Practice writing Coordinate Geometry Proofs1. 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.2. Given ABC with vertices A(-4,2), B(4,4) and C(2,-6), the midpoints of AB and BC are Pand Q, respectively, and PQ is drawn. Prove by coordinate geometry:a. PQ ACb. PQ ½ AC29

3. Quadrilateral ABCD has vertices A(-6,3), B(-3,6), C(9,6) and D(-5,-8). Prove thatquadrilateral ABCD is:c. a trapezoidd. not an isosceles trapezoid4. The vertices of quadrilateral ABCD are A(-3,-1), B(6,2), C(5,5) and D(-4,2). Prove thatquadrilateral ABCD is a rectangle.30

5. The vertices of quadrilateral ABCD are A(-3,1), B(1,4), C(4,0) and D(0,-3). Prove thatquadrilateral ABCD is a square.6. Quadrilateral METS has vertices M(-5, -2), E(-5,3), T(4,6) and S(7,2). Prove by coordinategeometry that quadrilateral METS is an isosceles trapezoid.31

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.