The Distance Formula (and mid point)

What is to be learned? How to calculate the distance between two points

x y A (3, 1) B (7, 1) AB = 4 units

x y A (3, 1) C (7, 4) ? 4 units

x y A (3, 1) C (7, 4) ? 4 units (7 – 3) (x 1, y 1 ) (x 2, y 2 ) (x 2 – x 1 ) 3 units (4 – 1) (y 2 – y 1 ) Pythagoras AC 2 = = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2

The Distance Formula AB 2 = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 For Points A(x 1, y 1 ) and B(x 2, y 2 ) orAB =√ (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 Distance between A (2, 8) and B (7, 5) (x 1, y 1 ) (x 2, y 2 ) AB =√ (7 – 2) 2 + (5 – 8) 2 AB =√ (-3) 2 = √34

The Distance Formula AB 2 = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 For Points A(x 1, y 1 ) and B(x 2, y 2 ) orAB =√ (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2

Ex Distance between C (2, -5) and D (4, 5) (x 1, y 1 ) (x 2, y 2 ) CD =√ (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 CD =√ (4 – 2) 2 + (5 – (-5)) 2 Careful! CD =√ = √104

What is to be learned? How to calculate the distance between two points How to use the distance formula for problem solving.

Distance Formula Applications Circle centre A (2, 3) has radius 5 units Is B (4, 7) inside outside or on circle? A (2, 3) B (4, 7) √20 5 Inside

Key Question A(1, 2) B(3, 0) C (-1, -2) are vertices of triangle ABC. Prove that it is isosceles. AC = √20 and BC = √20 Therefore AB = BC, so triangle is isosceles.

Mid Points x y A (3, 1) B (9, 5) (, ) Midway between 3 and 9 6 Midway between 1 and

Mid Point of a Line For Points A(x 1, y 1 ) and B(x 2, y 2 ) Mid Point = x 1 + x 2, y 1 + y 2 2 ( ) or use common sense!

Ex Mid Point EF E(2, -5) and F (-6, 8) (x 1, y 1 ) (x 2, y 2 ) Mid Point = 2 + (-6), ( ) 2 = -4, 3 ( ) 2 = (-2, 1½ )

Is this right angled? = = = 144 no

Is this right angled? If right angled triangle (if rt. L ) 13 2 = LHS13 2 = 169 RHS = 169 LHS = RHS → rt. L