Copyright © Cengage Learning. All rights reserved. 9 Radicals and Rational Exponents Copyright © Cengage Learning. All rights reserved.

9.2 Applications of the Pythagorean Theorem and the Distance Formula Section Applications of the Pythagorean Theorem and the Distance Formula 9.2 Copyright © Cengage Learning. All rights reserved.

Objectives Apply the Pythagorean theorem to find the length of one side of a right triangle. Find the distance between two points on the coordinate plane. 1 2

Applications of the Pythagorean Theorem and the Distance Formula In this section, we will discuss the Pythagorean theorem, a theorem that shows the relationship of the sides of a right triangle. We will then use this theorem to develop a formula to calculate the distance between two points on the coordinate plane.

Apply the Pythagorean theorem to find Apply the Pythagorean theorem to find the length of one side of a right triangle 1.

Apply the Pythagorean theorem to find the length of one side of a right triangle If we know the lengths of the two legs of a right triangle, we can find the length of the hypotenuse (the side opposite the 90 angle) by using the Pythagorean theorem. In fact, if we know the lengths of any two sides of a right triangle, we can find the length of the third side. Pythagorean Theorem If a and b are the lengths of two legs of a right triangle and c is the length of the hypotenuse, then a2 + b2 = c2

Apply the Pythagorean theorem to find the length of one side of a right triangle In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs. Suppose the right triangle shown in Figure 9-6 has legs of length 3 and 4 units. Figure 9-6

Apply the Pythagorean theorem to find the length of one side of a right triangle To find the length of the hypotenuse, we can use the Pythagorean theorem. a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 To find the value of c, we ask “what number when squared is equal to 25?” There are two such numbers: the positive square root of 25 and the negative square root of 25. Substitute. Simplify. Add.

Apply the Pythagorean theorem to find the length of one side of a right triangle Since c represents the length of the hypotenuse and cannot be negative, it follows that c is the positive square root of 25. 5 = c The length of the hypotenuse is 5 units. Recall that the radical symbol represents the positive, or principal, square root of a number.

Example 1 – Fighting Fires To fight a forest fire, the forestry department plans to clear a rectangular fire break around the fire, as shown in Figure 9-7. Crews are equipped with mobile communications with a 3,000-yard range. Can crews at points A and B remain in radio contact? Figure 9-7

Example 1 – Solution Points A, B, and C form a right triangle. The lengths of its sides are represented as a, b, and c in yards, where a is opposite point A, b is opposite point B, and c is opposite point C. To find the distance c, we can use the Pythagorean theorem, substituting 2,400 for a and 1,000 for b and solving for c. a2 + b2 = c2 2,4002 + 1,0002 = c2 5,760,000 + 1,000,000 = c2 Substitute. Square each value.

Example 1 – Solution cont’d 6,760,000 = c2 2,600 = c The two crews are 2,600 yards apart. Because this distance is less than the 3,000-yard range of the radios, they can communicate. Add. Since c represents a length, it must be the positive square root of 6,760,000. Use a calculator to find the square root.

Find the distance between two points on the coordinate plane 2.

Find the distance between two points on the coordinate plane We can use the Pythagorean theorem to develop a formula to find the distance between any two points that are graphed on a rectangular coordinate system. To find the distance d between points P and Q shown in Figure 9-8, we construct the right triangle PRQ. Figure 9-8

Find the distance between two points on the coordinate plane Because line segment RQ is vertical, point R will have the same x-coordinate as point Q. Because line segment PR is horizontal, point R will have same y-coordinate as point P. The distance between P and R is | x2 – x1 |, and the distance between R and Q is | y2 – y1 |.

Find the distance between two points on the coordinate plane We apply the Pythagorean theorem to the right triangle PRQ to get (PQ)2 = (PR)2 + (RQ)2 d2 = | x2 – x1 |2 + | y2 – y1 |2 d2 = (x2 – x1)2 + (y2 – y1)2 Read PQ as “the length of segment PQ.” Substitute the value of each expression. | x2 – x1 |2 = (x2 – x1)2 and | y2 – y1 |2 = (y2 – y1)2

Find the distance between two points on the coordinate plane (1) Equation 1 is called the distance formula. Distance Formula The distance d between two points (x1, y1) and (x2, y2) is given by the formula Since d represents a length, it must be the positive square root of (x2 – x1)2 + (y2 – y1)2.

Example 2 Find the distance between the points (–2, 3) and (4, –5). Solution: To find the distance, we can use the distance formula by substituting 4 for x2, –2 for x1, –5 for y2, and 3 for y1 . Substitute. Simplify.

Example 2 – Solution The distance between the two points is 10 units. cont’d The distance between the two points is 10 units. Simplify. Square each value. Add. Take the square root.