1 Then the lengths of the legs of ABC are: AC = |4 – (–3)| = |7| = 7 BC = |6 – 2| = |4| = 4 To find the distance between points A and B, draw a right triangle as shown. Points A and C lie on a horizontal line. The distance between points A and C is the absolute value of the difference of their x - coordinates. Points B and C lie on a vertical line. The distance between points B and C is the absolute value of the difference of their y -coordinates. The Distance and Midpoint Formulas 9.5 LESSON

2 To find the distance between points A and B, draw a right triangle as shown. Points A and C lie on a horizontal line. The distance between points A and C is the absolute value of the difference of their x - coordinates. Points B and C lie on a vertical line. The distance between points B and C is the absolute value of the difference of their y -coordinates. AC = 7, BC = 4 Because AB is the length of the hypotenuse of the triangle, use the Pythagorean theorem to find AB. Pythagorean theorem Take positive square root of each side. Substitute 7 for AC and 4 for BC. Simplify. (AB) 2 = (AC) 2 + (BC) 2 AB = √ (AC) 2 + (BC) 2 = √ = √ 65 The example above suggests the following formula for finding the distance between any two points in a plane. The Distance and Midpoint Formulas 9.5 LESSON

3 The Distance Formula The distance between two points in a coordinate plane is equal to the square root of the sum of the horizontal change squared and the vertical change squared. Words Algebra d = √ (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 The Distance and Midpoint Formulas 9.5 LESSON

4 EXAMPLE 1 Finding the Distance Between Two Points Find the distance between the points M(6, 3) and N(5, 7). Distance formula Substitute 5 for x 2, 6 for x 1, 7 for y 2, and 3 for y 1. Subtract. Evaluate powers. Add. ANSWER d = √ (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 = √ (5 – 6) 2 + (7 – 3) 2= √ (5 – 6) 2 + (7 – 3) 2 = √ (–1) = √ = √ 17 The distance between the points M(6, 3) and N(5, 7) is √ 17 units. The Distance and Midpoint Formulas 9.5 LESSON

5 EXAMPLE 2 Using the Distance Formula Bird Watching A bird watcher uses a grid to photograph and record the locations of birds feeding on the ground. The grid is made of fishing net with strands that are 1 foot apart. In the grid shown, each point represents the location of a bird. How far apart are the birds at points A and B ? SOLUTION The coordinates of point A are (3, 2). The coordinates of point B are (7, 5). Distance formula Substitute 7 for x 2, 3 for x 1, 5 for y 2, and 2 for y 1. Subtract. Evaluate powers. d = √ (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 = √ (7 – 3) 2 + (5 – 2) 2= √ (7 – 3) 2 + (5 – 2) 2 = √ = √ = √ The Distance and Midpoint Formulas 9.5 LESSON

6 EXAMPLE 2 Using the Distance Formula Bird Watching A bird watcher uses a grid to photograph and record the locations of birds feeding on the ground. The grid is made of fishing net with strands that are 1 foot apart. In the grid shown, each point represents the location of a bird. How far apart are the birds at points A and B ? SOLUTION The coordinates of point A are (3, 2). The coordinates of point B are (7, 5). Add. Simplify. = √ 25 = 5= 5 = √ ANSWER The birds at points A and B are 5 feet apart. The Distance and Midpoint Formulas 9.5 LESSON

7 Midpoint The midpoint of a segment is the point on the segment that is equally distant from the endpoints. The Midpoint Formula The coordinates of the midpoint of a segment are the average of the endpoints’ x -coordinates and the average of the endpoints’ y -coordinates. Words Algebra M = x 1 + x 2 2 y 1 + y 2 2, The Distance and Midpoint Formulas 9.5 LESSON

8 EXAMPLE 3 Finding a Midpoint Find the midpoint M of the segment with endpoints (3, 8) and (–9, –4). M = x 1 + x 2 2 y 1 + y 2 2, Midpoint formula Substitute 3 for x 1, –9 for x 2, 8 for y 1, and –4 for y 2. Simplify. = 3 + (–9) (–4) 2, = (–3, 2) The Distance and Midpoint Formulas 9.5 LESSON

9 Slope If points A(x 1, y 1 ) and B(x 2, y 2 ) do not lie on a vertical line, you can use coordinate notation to write a formula for the slope of the line through A and B. slope = y 2 – y 1 x 2 – x 1 = difference of y -coordinates difference of x -coordinates The Distance and Midpoint Formulas 9.5 LESSON

10 EXAMPLE 4 Finding Slope Find the slope of the line through (3, 7) and (8, –3). Slope formula Substitute –3 for y 2, 7 for y 1, 8 for x 2, and 3 for x 1. Simplify. slope = y 2 – y 1x 2 – x 1y 2 – y 1x 2 – x 1 = –3 – 7 8 – 3 = –2 = –10 5 The Distance and Midpoint Formulas 9.5 LESSON