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Holt McDougal Algebra The Midpoint and Distance Formulas Apply the formula for midpoint. Use the distance formula to find the distance between two points. Objective

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Holt McDougal Algebra The Midpoint and Distance Formulas midpoint Vocabulary

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Holt McDougal Algebra The Midpoint and Distance Formulas 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.

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Holt McDougal Algebra The Midpoint and Distance Formulas

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

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Holt McDougal Algebra The Midpoint and Distance Formulas 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. Substitute. Simplify. E(–2, 3) F(5, –3)

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

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Holt McDougal Algebra The Midpoint and Distance Formulas Additional Example 2 Continued Multiply both sides by 2. Isolate the variables. –2 = –5 + x +5 3 = x 6 = 4 + y 4 Simplify. 2 = y Set the coordinates equal. Step 3 Find the x-coordinate. Find the y-coordinate. Simplify.

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

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Holt McDougal Algebra The Midpoint and Distance Formulas 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.

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Holt McDougal Algebra The Midpoint and Distance Formulas Check It Out! Example 2 Continued Multiply both sides by 2. Simplify. Isolate the variables. –2 = –6 + x +6 2 = –1 + y +1 4 = x3 = y Set the coordinates equal. Step 3 Find the x-coordinate. Find the y-coordinate.

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

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Holt McDougal Algebra The Midpoint and Distance Formulas 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.

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Holt McDougal Algebra The Midpoint and Distance Formulas The Pythagorean Theorem states that if a right triangle has legs of lengths a and b and a hypotenuse of length c, then a 2 + b 2 = c 2. Remember!

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Holt McDougal Algebra The Midpoint and Distance Formulas 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 (x 1, y 1 ) and (3, –2) for (x 2, y 2 ). Subtract. Simplify powers. Add. Find the square root to the nearest hundredth.

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Holt McDougal Algebra The Midpoint and Distance Formulas Additional Example 3 Continued A (–2, –2) B (4, 3) 6 5 Use the Distance Formula to find the distance, to the nearest hundredth, from A(–2, –2) to B(4, 3).

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Holt McDougal Algebra The Midpoint and Distance Formulas 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 (x 1, y 1 ) and (-3, -1) for (x 2, y 2 ). Add. Simplify powers. Add. Find the square root to the nearest hundredth.

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Holt McDougal Algebra The Midpoint and Distance Formulas Check It Out! Example 3 Continued S(–3, –1) R(3, 2) 6 3 Use the Distance Formula to find the distance, to the nearest tenth, from R(3, 2) to S(–3, –1).

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Holt McDougal Algebra The Midpoint and Distance Formulas 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? Additional Example 4: Application Substitute. Add. Simplify powers. Find the square root to the nearest tenth. It is or meters from the roller coaster to Ferris Wheel.

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Holt McDougal Algebra The Midpoint and Distance Formulas 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? d 17.7 miles Substitute. Square. Simplify powers. Find the square root to the nearest tenth.

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