Coordinate Proofs Lesson 6-2

Warm-up Are the following lines perpendicular, parallel, or neither? y = 2x + 5 3. y = 2x + 6 -2x + y = 3 3x + 6y = 12 y = 3x + 4 4. 4x + 2y = 8 9x + 3y = 15 y = -2x + 4

Refresher Are the following lines perpendicular, parallel, or neither? (2, 4) and (3, 6) 3. (5, 7) and (2, 4) (0, 0) and (2, 1) (3, 5) and (1, 7) (4, 1) and (3, 5) 4. (3, 5) and (8, 5) (1, 2) and (5, 3) (2, 7) and (2, -4)

Midpoint Formula To calculate the midpoint between two points, you average the x’s to get the x-value and average the y’s to get the y-value. The formula is Example: Find the midpoint of (5, 3) and (1, -5).

Distance Formula To calculate the distance between two points, we treat the distance between them as the hypotenuse of a right triangle. The difference in x’s is one side of the triangle and the difference in y’s is the other side. The formula is Example: Find the distance between (3, 8) and (2, 4).

Let’s Practice What is the distance from (2, 4) to (5, - 3)? What is the midpoint of (2, -1) and (-4, 3) The distance from point (1, 1) to another point (x, 5) is 5 units. What is x? The midpoint of (3, -2) and another point (x, y) is (-2, 5). What is the point (x, y)?

Classifying Triangles Scalene: No sides have the same measure Isosceles: Two sides have the same measure Equilateral: Three sides have the same measure

Coordinate Proofs Tell if a triangle with vertices at (-3,2), (-2,-2), and (1,-2) is a scalene, isosceles, or equilateral triangle.

First Step: Plot the points and draw the triangle. B: (-2,-2) C: (1, -2) c b

For side AB use (-3,2) and (-2,-2) Step Two: Use the distance formula to find the measures of the three sides For side AB use (-3,2) and (-2,-2) √(-3 - -2)2 + (2 - -2)2 √(-1)2 + (4)2 √1 + 16 √17 4.1

For side BC use (-2,-2) and (1,-2) Step Two: Use the distance formula to find the measures of the three sides For side BC use (-2,-2) and (1,-2) √(-2 - 1)2 + (-2 - -2)2 √(-3)2 + (0)2 √9 + 0 √9 3

For side AC use (-3,2) and (1,-2) Step Two: Use the distance formula to find the measures of the three sides For side AC use (-3,2) and (1,-2) √(-3 - 1)2 + (2 - -2)2 √(-4)2 + (4)2 √16 + 16 √32 5.7

Step Three Use your information AB = 4.1 BC = 3 AC = 5.7 No side measures the same The triangle is: Scalene

Coordinate Proofs Tell if a triangle with vertices at (-5,-3), (7,-3), and (1,5) is a scalene, isosceles, or equilateral triangle.

First Step: Plot the points and draw the triangle. b: (-5,-3) c: (7, -3) b c

For side AB use (1,5) and (-5,-3) Step Two: Use the distance formula to find the measures of the three sides For side AB use (1,5) and (-5,-3) √(1 - -5)2 + (5 - -3)2 √(6)2 + (8)2 √36 + 64 √100 10

For side BC use (-5,-3) and (7,-3) Step Two: Use the distance formula to find the measures of the three sides For side BC use (-5,-3) and (7,-3) √(-5 - 7)2 + (-3 - -3)2 √(-12)2 + (0)2 √144 + 0 √144 12

For side AC use (1,5) and (7,-3) Step Two: Use the distance formula to find the measures of the three sides For side AC use (1,5) and (7,-3) √(1 - 7)2 + (5 - -3)2 √(6)2 + (8)2 √36 + 64 √100 10

Step Three Use your information AB = 10 BC = 12 AC = 10 Two sides measures the same The triangle is: Isosceles

Tell if a quadrilateral with vertices at Coordinate Proofs Tell if a quadrilateral with vertices at A: (-1,4) B: (-1,-2) C: (5,-2) D: (5,4) is square

First Step: Plot the points and draw the triangle. b: (-1,-2) c: (5,-2) d: (5,4) c b

For side AB use (-1,4) and (-1,-2) Step Two: Use the distance formula to find the measures of the three sides For side AB use (-1,4) and (-1,-2) √(-1 - -1)2 + (4 - -2)2 √(0)2 + (6)2 √0 + 36 √36 6

For side BC use (-1,-2) and (5,-2) Step Two: Use the distance formula to find the measures of the three sides For side BC use (-1,-2) and (5,-2) √(-1 - 5)2 + (-2 - -2)2 √(-6)2 + (0)2 √36 + 0 √36 6

For side CD use (5,-2) and (5,4) Step Two: Use the distance formula to find the measures of the three sides For side CD use (5,-2) and (5,4) √(5 - 5)2 + (-2 - 4)2 √(0)2 + (-6)2 √0 + 36 √36 6

For side AD use (-1,4) and (5,4) Step Two: Use the distance formula to find the measures of the three sides For side AD use (-1,4) and (5,4) √(-1 - 5)2 + (4 - 4)2 √(-6)2 + (0)2 √36 + 0 √36 6

Step Three Use your information AB = 6 BC = 6 CD = 6 AD = 6 All sides measures the same The quadrilateral is: A Rhombus How would we prove it’s a square?