Chapter 11, Section 3: Distance and Midpoint Formulas

1. Warm Up: Find the missing length to the nearest tenth of a unit. Both triangles are right triangles. Triangle 1: Legs: 8ft and 12ft; find Hypotenuse. Triangle 2: Leg: 10mm, Hypotenuse: 25mm; find Leg.

2. Distance and Midpoint Formulas Chapter 11, Section 3

3. Finding Distance Use Pythagorean Theorem to find the length of a segment on a coordinate plane. Make a Right Triangle to do this. Or, just use the Distance Formula that is based off of Pythagorean's Theorem. Distance = √ (x ₂ – x ₁ ) ² + (y ₂ – y ₁ ) ² X and Y are from coordinate points. ex. (5, -2)

4. Find the Distance between A( 6 , 3 ) and B( 1 , 9 ) D = √ ( x ₂ – x ₁ ) ² + ( y ₂ – y ₁ ) ² It doesn't matter which coordinate is 1 or 2. Because a -#² = +# D = √ ( 6 ₂ – 1 ₁ ) ² + ( 9 ₂ – 3 ₁ ) ² D = √ ( 5 ) ² + ( 6 ) ² D = √ ( 25 + 36 ) D = √ (61) D ≈ 7.8 (rounded to tenth)

5. Use Distance Formula D = √ ( x ₂ – x ₁ ) ² + ( y ₂ – y ₁ ) ² Distance 1: ( 3 , 8 ), ( 2 , 4 ) Distance 2: ( 10 , -3 ), ( 1 , 0 )

6. Use Distance Formula to Determine Perimeter Find Distance between each point, then add them to find perimeter. AB = ? BC = ? CD = ? DA = ? D (3, 3) A (0, -1) B (8, 0) C (9, 4) √ 65 √ 17 √ 37 √ 25 = 5 These numbers add up to 23.2681259 units, which is the perimeter.

7. Midpoint Formula The midpoint of a segment is the POINT M. The midpoint is a dot with a coordinate (x, y). M = ( [x ₁ + x₂]/2 , [y₁ + y₂]/2 ) Take the x coordinates, add, divide by 2 = new x coordinate. Take the y coordinates, add, divide by 2 = new y coordinate. M = ( x , y )

8. Find the Midpoint M = ( [x ₁ + x₂]/2 , [y₁ + y₂]/2 ) Find the midpoint between: G( -3 , 2 ) and H( 7 , -2 ) ( [ -3 + 7 ]/2, [ 2 + -2 ]/2 ) ( [4]/2, [0]/2 ) ( 2, 0 ) ← Midpoint between G and H

9. Find the Midpoints Midpoint between A(2, 5) and B(8, 1): Midpoint between P(-4, -2) and Q(2, 3):

10. Assignment #32 Pages 575-576: 1-6 all, 8-21 all.