2. Aim: Distance Formula Course: Applied Geometry Do Now: Aim: How do we use the Pythagorean Theorem to find the distance between two points? In inches, the lengths of the diagonals of a rhombus are 30 and 40. Find the length of one side of the rhombus.

3. Aim: Distance Formula Course: Applied Geometry Pythagorean Theorem PR Q ? Q ? PR 4 3 What is the measure of the hypotenuse of the right  PRQ? Find the measure of the hypotenuse of a right triangle with legs of 3 and 4. 1. c 2 = a 2 + b 2 Pythagorean Theorem 2. c 2 = 3 2 + 4 2 = 9 + 16 = 25 3. If c 2 = 25, then c = 5. 5 5

4. Aim: Distance Formula Course: Applied Geometry Pythagorean Theorem Find the measure of the hypotenuse of a right triangle with legs of 5 and 12. 1. c 2 = a 2 + b 2 Pythagorean Theorem 2. c 2 = 5 2 + 12 2 = 25 + 144 = 169 3. If c 2 = 169, then c = 13. What is the measure of the hypotenuse of right  ABC? AB C 12 5 13

5. Aim: Distance Formula Course: Applied Geometry Developing the Distance Formula What is the length of RS? What is the length of RT? What is the length of ST? R T S 8 4 1. c 2 = a 2 + b 2 Pythagorean Theorem 3. ST 2 = 8 2 + 4 2 = 64 +16 = 80 4. If ST 2 = 80, then ST = 2. ST 2 = RS 2 + RT 2 Pythagorean Theorem 8 4

6. Aim: Distance Formula Course: Applied Geometry Developing the Distance Formula Plot the following coordinate points: A(1, 1), B(4, 6) and connect them. How do we find the length of AB? 5. (AB) 2 = 3 2 + 5 2 = 9 + 25 = 34 6. If (AB) 2 = 34, then AB = 4. (AB) 2 = (AC) 2 + (BC) 2 Pythagorean Thr. 2. AC = (x C - x A ) 3. BC = (y B - y C ) 1. AC = (4 - 1) = 3 A (1,1) C(4,1) (4,6)B 3 5 BC = (6 - 1) = 5

7. Aim: Distance Formula Course: Applied Geometry A (-2,1) B(4,-2) C (-2,-2) 3 6 Developing the Distance Formula Plot the following coordinate points: A(-2,1), B(4,-2) and connect them. What is the length of AB? 4. (AB) 2 = 3 2 + 6 2 = 9 + 36 = 45 5. If (AB) 2 = 45, then AB = 1. Length of AC = (y A - y C ) 2. Length of BC = (x B - x C ) AC = (1 - -2) = 3 BC = (4 - -2) = 6 Is distance a negative number? NO!! 3. (AB) 2 = (AC) 2 + (BC) 2 Pyth. Theorem

8. Aim: Distance Formula Course: Applied Geometry Developing the Distance Formula (x 2, y 2 ), (x 1, y 1 ) are endpoints of a line segment. What is the length d of this line segment? 1. Length of a = (x 2 - x 1 ) 2. Length of b = (y 2 - y 1 ) 3. (d) 2 = (a) 2 + (b) 2 Pythagorean Theorem (x 2, y 2 ) (x 1, y 1 ) (x 2, y 1 ) 4. (d) 2 = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 d a b Distance Formula 5. d

9. Aim: Distance Formula Course: Applied Geometry The Distance Formula (x 2, y 2 ) are the coordinates of one point and (x 1, y 1 ) are the coordinates of the second point. Ex: Find the distance between the points (3,-1) and (-2, 3). To find the distance between two points on the coordinate plane we use the distance formula. The distance formula is based on the Pythagorean Theorem.

10. Aim: Distance Formula Course: Applied Geometry Model Problem Use the distance formula to find the distance between points A(-2, 7) and B(1, 2). A(-2,7) B(1,2) (x 2, y 2 ) (x 1, y 1 ) d =d =

11. Aim: Distance Formula Course: Applied Geometry Model Problem Given rectangle ABCD with vertices A(8, 3), B(8, 9), and C(1, 9), and D(1, 3). Show the diagonals of the rectangle are congruent. A(8,3) B(8,9), C(1,9) D(1,3) Show BD  AC BD  AC

12. Aim: Distance Formula Course: Applied Geometry Model Problems Which point lies furthest from the origin? 1) (0,-9) 2) (-7,6) 3) (8,5) 4) (-2,9) The coordinates of the opposite vertices of quadrilateral ABCD are A(0, 0) and C(k, 0). If AC = 10, find the positive values of k.

13. Aim: Distance Formula Course: Applied Geometry The Product Rule