Objective Apply the formula for midpoint. Use the distance formula to find the distance between two points.

In Lesson 5-4, you used the coordinates of points to determine the slope of lines. You can also use coordinates to determine the midpoint of a line segment on the coordinate plane. The midpoint of a line segment is the point that divides the segment into two congruent segments. Congruent segments are segments that have the same length. You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints.

Additional Example 1: Finding the Coordinates of a Midpoint Find the coordinates of the midpoint of GH with endpoints G(–4, 3) and H(6, –2). Write the formula. G(–4, 3) Substitute. H(6, -2) Simplify.

Check It Out! Example 1 Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3). Write the formula. E(–2, 3) Substitute. F(5, –3) Simplify.

Additional Example 2: Finding the Coordinates of an Endpoint P is the midpoint of NQ. N has coordinates (–5, 4), and P has coordinates (–1, 3). Find the coordinates of Q. Step 1 Let the coordinates of P equal (x, y). Step 2 Use the Midpoint Formula.

Additional Example 2 Continued Step 3 Find the x-coordinate. Find the y-coordinate. Set the coordinates equal. Multiply both sides by 2. –2 = –5 + x Simplify. 6 = 4 + y Isolate the variables. +5 +5 −4 −4 3 = x Simplify. 2 = y

Additional Example 2 Continued The coordinates of Q are (3, 2). Check Graph points Q and N and midpoint P. N (–5, 4) P(–1, 3) Q (3, 2)

Check It Out! Example 2 S is the midpoint of RT. R has coordinates (–6, –1), and S has coordinates (–1, 1) . Find the coordinates of T. Step 1 Let the coordinates of T equal (x, y) . Step 2 Use the Midpoint Formula.

Check It Out! Example 2 Continued Step 3 Find the x-coordinate. Find the y-coordinate. Set the coordinates equal. Multiply both sides by 2. –2 = –6 + x Simplify. 2 = –1 + y +6 +6 Isolate the variables. +1 +1 4 = x Simplify. 3 = y

Check It Out! Example 2 Continued The coordinates of T are (4, 3) Check Graph points R and S and midpoint T. T(4, 3) S(–1, 1) R(–6, –1)

You can also use coordinates to find the distance between two points or the length of a line segment. To find the length of segment PQ, draw a horizontal segment from P and a vertical segment from Q to form a right triangle.

The Pythagorean Theorem states that if a right triangle has legs of lengths a and b and a hypotenuse of length c, then a2 + b2 = c2. Remember!

Additional Example 3: Finding Distance in the Coordinate Plane Use the Distance Formula to find the distance, to the nearest hundredth, from A(–2, –2) to B(4, 3). Distance Formula Substitute (4, –2) for (x1, y1) and (3, –2) for (x2, y2). Subtract. Simplify powers. Add. Find the square root to the nearest hundredth.

Additional Example 3 Continued Use the Distance Formula to find the distance, to the nearest hundredth, from A(–2, –2) to B(4, 3). 6 B (4, 3) 5 A (–2, –2)

Check It Out! Example 3 Use the Distance Formula to find the distance, to the nearest tenth, from R(3, 2) to S(–3, –1). Distance Formula Substitute (3, 2) for (x1, y1) and (-3, -1) for (x2, y2). Add. Simplify powers. Add. Find the square root to the nearest hundredth.

Check It Out! Example 3 Continued Use the Distance Formula to find the distance, to the nearest tenth, from R(3, 2) to S(–3, –1). R(3, 2) 6 3 S(–3, –1)

Additional Example 4: Application Each unit on the map represents 100 meters. To the nearest tenth of a meter, how far is it from the roller coaster to the Ferris wheel? Substitute. It is 7.211  100 or 721.1 meters from the roller coaster to Ferris Wheel. Add. Simplify powers. Find the square root to the nearest tenth.

Check It Out! Example 4 Jacob takes a boat from Pahokee to Clewiston. To the nearest tenth of a mile, how far does he travel? Substitute. Square. Simplify powers. Find the square root to the nearest tenth. d  17.7 miles

Lesson Quiz: Part I 1. Find the coordinates of the midpoint of AB with endpoints A(–1, 4) and B( 4, –3). 2. M is the midpoint of AB. A has coordinates (–5, 7), and M has coordinates (–1, 8). Find the coordinates of B. (3, 9) 3. Use the Distance Formula to find the distance, to the nearest hundredth, from A (–1, –3) to B (7, –1). 8.25

Lesson Quiz: Part II 4. Each unit on the map represents 1 meter. To the nearest tenth of a meter, what is the distance from the fountain to the statue? 18.0 m