The Distance and Midpoint Formulas! To be used when you want to find the distance between two points or the midpoint between two points

You have learned… 1 The Pythagorean a cTheorem a 2 + b 2 = c 2 a 2 + b 2 = c 2 b 2 When we are finding c we are really finding the distance between angles 1 and 2 If we solve for c we get c = √(a 2 + b 2 ) This is how the distance formula is derived – it is useful when we know coordinates

The Distance Formula! The distance formula is d = √((y 2 – y 1 ) 2 + (x 2 – x 1 ) 2 ) And is used to find the distance between two points on the coordinate plane Let’s practice one!

Example: What is the distance between (2, -6) and (-3, 6)? First, identify x1, y1, x2 and y2 x 1 = 2y 1 = -6 x 2 = -3y 2 = 6 Now, we use the formula: d = √((6 – -6) 2 + (-3 – 2) 2 ) = √( (-5 2) ) d = 13

The Midpoint Formula! The midpoint formula is used to find the coordinate that is the exact midpoint between two other coordinates The x-coordinate of the midpoint is found by (x 2 + x 1 )/2 The y-coordinate of the midpoint is found by (y 2 + y 1 )/2 So, the coordinate of the midpoint is: (x 2 + x 1 )/2, (y 2 + y 1 )/2

Example: What is the coordinate of the midpoint between (1, 2) and (-5, 6)? Again, identify x1, y1, x2 and y2 x 1 = 1y 1 = 2 x 2 = -5y 2 = 6 Now use the formula: x-coordinate: (1 + -5)/2 = -2 y-coordinate: (2 + 6)/2 = 8 So, the midpoint is located at the coordinate (-2, 8)

Another Example On the coordinate plane, it is given that the midpoint of points A and B is (5, 7). If point A is located at (-1, 2), where is point B located? In this case, we know the midpoint and the coordinate of point A. In a sense, we need to work backwards. Let’s define what we have:

x 1 = -1y 1 = 2 x 2 = ?y 2 = ? So we know… (-1 + x 2 )/2 = 5(2 + y 2 )/2 = x 2 = 10(2 + y 2 ) = 14 x 2 = 11y 2 = 12 So the coordinate of point B is (11, 12)