Section P.6: The Distance and Midpoint Formulas Formula for finding the Distance between two points on the xy-coordinate plane: Let (𝑥1,𝑦1) and (𝑥2,𝑦2) be the coordinates of two points on the 𝑥𝑦 plane. The distance 𝑑 between them equals: 𝑑= 𝑥2−𝑥1 2 + 𝑦2−𝑦1 2

Example of Using the Distance Formula Find the distance between the points (−1, −4) and (3, −10). Let 𝑥1,𝑦1 =(−1,−4) and let 𝑥2,𝑦2 = 3,−10 Plug into the formula: 𝑑= 𝑥2−𝑥1 2 + 𝑦2−𝑦1 2 = 3− −1 2 + −10− −4 2 = 3+1 2 + −10+4 2 = 4 2 + −6 2 = 16+36 = 52 =2 13

Verifying a Right Triangle Show that the points (2,1), (4,0), and (5,7) are all vertices of a right triangle. To do this, I would advise you to draw a graph of the three points and draw three line segments connecting the points. You need to find the length of each line segment as follows: Find the distance from (2,1) to (4,0) Find the distance from (4,0) to (5,7) Find the distance from (5,7) to (2,1).

Pythagorean Theorem To be a right triangle, the square of the longest side must equal the sum of the squares of the two shorter sides. 𝑎 2 + 𝑏 2 = 𝑐 2

Midpoint Formula Suppose (𝑥1,𝑦1) and (𝑥2,𝑦2) are endpoints of a line segment. To find the midpoint 𝑚 between the two endpoints, use this formula: 𝑚=( 𝑥1+𝑥2 2 , 𝑦1+𝑦2 2 ) The average of the x coordinates of the endpoints becomes the x coordinate of the midpoint, while the average of the y coordinates of the endpoints becomes the y coordinate of the midpoint.

Example: Using the Midpoint Formula Find the midpoint between (−3,4) and 1,7 . Let 𝑥1,𝑦1 =(−3,4) and let 𝑥2,𝑦2 = 1,7 𝑚= 𝑥1+𝑥2 2 , 𝑦1+𝑦2 2 = −3+1 2 , 4+7 2 𝑚=( −2 2 , 11 2 )=(−1, 5.5)