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The Distance and Midpoint Formulas!

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 c Theorem a 2 + b 2 = c 2 b 2

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The Distance and Midpoint Formulas!

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

  2. You have learned… 1 The Pythagorean a c Theorem a2 + b2 = c2 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 = √(a2 + b2) This is how the distance formula is derived – it is useful when we know coordinates

  3. The Distance Formula! The distance formula is d = √((y2 – y1)2 + (x2 – x1)2) And is used to find the distance between two points on the coordinate plane Let’s practice one!

  4. Example: What is the distance between (2, -6) and (-3, 6)? First, identify x1, y1, x2 and y2 x1 = 2 y1 = -6 x2 = -3 y2 = 6 Now, we use the formula: d = √((6 – -6)2 + (-3 – 2)2) = √(122 + (-52)) d = 13

  5. 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 (x2 + x1)/2 The y-coordinate of the midpoint is found by (y2 + y1)/2 So, the coordinate of the midpoint is: (x2 + x1)/2, (y2 + y1)/2

  6. Example: What is the coordinate of the midpoint between (1, 2) and (-5, 6)? Again, identify x1, y1, x2 and y2 x1 = 1 y1 = 2 x2 = -5 y2 = 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)

  7. 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:

  8. x1 = -1 y1 = 2 x2 = ? y2 = ? So we know… (-1 + x2)/2 = 5 (2 + y2)/2 = 7 -1 + x2 = 10 (2 + y2) = 14 x2 = 11 y2 = 12 So the coordinate of point B is (11, 12)

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