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

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.

Lines With the same midpoint bisect each other

Midpoint Formula:

xx1  2 yy12 mid  , 22

Distance: We use distance to show line segments are equal.

You can use the Pythagorean Theorem or the formula:

2 2 d  (x)  (y)

1

Day 1 – Using Coordinate Geometry To Prove Right Triangles and Parallelograms

Warm – Up

Explaination:______

Explaination:______

2

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:

d  (x)2  (y)2

Work: Calculate the Distances of all three sides to show Pythagorean’s Theorem is true.

a2 + b2 = c2

( )2 + ( )2 = ( )2

Statement:

3

“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:

2 2 d  (x)  (y)

Work: Calculate the Distances of all three sides to show Pythagorean’s Theorem is true.

 _____  _____

a2 + b2 = c2

( )2 + ( )2 = ( )2

Statement:

4

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.

Show:

Formula:

d  (x)2  (y)2

Work: 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 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

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

7

SUMMARY

Proving Right Triangles

Proving Parallelograms

Exit Ticket

8

Homework

1.

2.

9

3.

4.

10

6. Prove that quadrilateral LEAP with the vertices L(-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 Squares

Warm – Up

1.

2.

13

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)2  (y)2

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 ______.

14

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: d  (x)2  (y)2

Work Step 1: Calculate the Distances of all four sides to show that all four sides equal.

 _____  _____  _____  _____

Statement:

 ______is a Rhombus because ______.

15

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: d  (x)2  (y)2

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 ______.

16

SUMMARY

Proving Rectangles Proving Rhombi

Proving Squares

17

Challenge

Exit Ticket

1.

2.

18

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.

19

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.

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 Trapezoids

Warm - Up

1.

2.

22

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 ______.

23

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 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:

d  (x)2  (y)2

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 ______.

24

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.

25

Challenge

SUMMARY

Exit Ticket

26

Day 3 - Homework

1.

2.

27

3. .

4. Triangle TOY has coordinates T(-4, 2), O(-2, -2), and Y(2, 0). Prove TOY is an isosceles right triangle.

28

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.

2. Given ABC with vertices A(-4,2), B(4,4) and C(2,-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

29

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(6,2), C(5,5) and D(-4,2). Prove that quadrilateral 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 that quadrilateral ABCD is a square.

6. Quadrilateral METS has vertices M(-5, -2), E(-5,3), T(4,6) and S(7,2). Prove by coordinate geometry that quadrilateral METS is an isosceles trapezoid.

31