Using the Pythagorean Theorem Lesson 3 3 2

Using the Pythagorean Theorem Lesson 3. 3. 2 1

Lesson 3. 3. 2 Using the Pythagorean Theorem California Standards: What it means for you: Measurement and Geometry 3. 2 You’ll see how to use the Pythagorean theorem to find missing side lengths of right triangles. Understand use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections. Measurement and Geometry 3. 3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. Key words: • • • Pythagorean theorem right triangle hypotenuse legs square root 2

Lesson 3. 3. 2 Using the Pythagorean Theorem In the last Lesson, you met the Pythagorean theorem and saw how it linked the lengths of the sides of a right triangle. Area = c 2 c b Area = b 2 a Area = a 2 In this Lesson, you’ll practice using theorem to work out missing side lengths in right triangles. 3

Lesson 3. 3. 2 Using the Pythagorean Theorem Use the Pythagorean Theorem to Find the Hypotenuse If you know the lengths of the two legs of a right triangle you can use them to find the length of the hypotenuse. c b a The theorem says that c 2 = a 2 + b 2 where c is the length of the hypotenuse, and a and b are the lengths of the two legs. So if you know the lengths of the legs you can put them into the equation, and solve it to find the length of the hypotenuse. 4

Lesson 3. 3. 2 Using the Pythagorean Theorem Example 1 Use the Pythagorean theorem to find the length of the hypotenuse of the right triangle shown. c cm 8 cm Solution c 2 = a 2 + b 2 First write out the equation c 2 = 6 2 + 8 2 Substitute in the side lengths that you know c 2 = 36 + 64 Simplify the expression 6 cm c 2 = 100 c= Take the square root of both sides c = 10 cm 5 Solution follows…

Lesson 3. 3. 2 Using the Pythagorean Theorem A lot of the time your solution won’t be a whole number. That’s because the last step of the work is taking a square root, which often leaves a decimal or an irrational number answer. c 2 = a 2 + b 2 c 2 = 6 2 + 8 2 c 2 = 36 + 64 c 2 = 100 c= c = 10 cm 6

Lesson Using the Pythagorean Theorem 3. 3. 2 Example 2 Use the Pythagorean theorem to find the length of the hypotenuse of the right triangle shown. c cm Solution 1 m c 2 = a 2 + b 2 First write out the equation c 2 = 1 2 + 1 2 Substitute in the side lengths that you know c 2 = 1 + 1 Simplify the expression c 2 = 2 c= m 1 m Cancel out the squaring by taking out the square root If you do this calculation on a calculator, you’ll see that is approximately equal to 1. 4 m. m 7 Solution follows…

Lesson 3. 3. 2 Using the Pythagorean Theorem The Pythagorean theorem is also useful for finding lengths on graphs that aren’t horizontal or vertical. y 4 A 3 B 2 1 0 0 1 2 3 4 5 x 8

Lesson 3. 3. 2 Example Using the Pythagorean Theorem 3 Find the length of the line segment KL. Solution Draw a horizontal and vertical line on the plane to make a right triangle. Now use the same method as before. y L 4 3 3 units 2 K 1 0 2 units 0 1 2 3 4 5 9 x Solution continues… follows…

Lesson Using the Pythagorean Theorem 3. 3. 2 Example 3 Find the length of the line segment KL. Solution (continued) KL 2 = a 2 + b 2 KL 2 = 32 + 22 =9+4 KL 2 = 13 KL = KL » 3. 6 units Write out the equation Substitute in the side lengths that you know y L 4 3 3 units 2 K 1 0 2 units 0 1 2 3 4 5 Simplify the expression Cancel out the squaring by taking the square root 10 x

Lesson 3. 3. 2 Using the Pythagorean Theorem Guided Practice Use the Pythagorean theorem to find the length of the hypotenuse in Exercises 1– 3. 1. 2. 8 units c ft 12 cm 5 ft c 2 = 122 + 52 c 2 = 144 + 25 c 2 = 169 c = 13 ft 15 units 3. 3. 6 cm c units c 2 = 152 + 82 c 2 = 225 + 64 c 2 = 289 c = 17 units c cm 1. 5 cm c 2 = 3. 62 + 1. 52 c 2 = 12. 96 + 2. 25 c 2 = 15. 21 c = 3. 9 cm 11 Solution follows…

Lesson 3. 3. 2 Using the Pythagorean Theorem Guided Practice 4. Use the Pythagorean theorem to find the length of the line segment XY. y 3 2 Y 1 0 XY 2 = 32 + 32 XY 2 = 9 + 9 XY 2 = 18 XY 2 = XY » 4. 2 units – 2 – 1 0 – 1 X – 2 x 1 2 12 Solution follows…

Lesson 3. 3. 2 Using the Pythagorean Theorem You Can Use the Theorem to Find a Leg Length If you know the length of the hypotenuse and one of the legs, you can use theorem to find the length of the “missing” leg. You just need to rearrange the formula: a 2 + b 2 = c 2 a 2 = c 2 – b 2 Subtract b 2 from both sides to get the a 2 term by itself. Remember that it doesn’t matter which of the legs you call a and which you call b. But the hypotenuse is always c. Now you can substitute in values to find the missing length as you did with the hypotenuse. 13

Lesson 3. 3. 2 Using the Pythagorean Theorem Example 4 Find the missing length in this right triangle. Solution 3 cm c 2 = a 2 + b 2 First write out the equation a 2 = c 2 – b 2 Rearrange it a 2 = cm a – 32 Substitute in the side lengths that you know a 2 = 58 – 9 Simplify the expression a 2 = 49 a= Take the square root of both sides a = 7 cm 14 Solution follows…

Lesson Using the Pythagorean Theorem 3. 3. 2 Guided Practice Use the Pythagorean theorem to calculate the missing lengths in Exercises 5– 8. 5. 6. 16 cm 20 cm a cm 7. a 2 = 202 – 162 a 2 = 400 – 256 a 2 = 144 a = 12 3. 4 ft 8. a units 10 units a ft 1. 6 ft a 2 = 136 – 102 a 2 = 136 – 100 a 2 = 36 a=6 units 5 units a 2 = 3. 42 – 1. 62 a 2 = 11. 56 – 2. 56 a 2 = 9 a=3 a units a 2 = 89 – 52 a 2 = 89 – 25 a 2 = 64 15 a. Solution =8 follows…

Lesson Using the Pythagorean Theorem 3. 3. 2 Independent Practice Use the Pythagorean theorem to find the value of c in Exercises 1– 5. 1. 2. 3. c cm 12 cm c = 15 0. 8 m cm c=1 cm 0. 6 m 9 cm 4. 5. c in 7 in c= 2 in 1. 5 cm 4. 8 m 3. 6 m c=6 1 cm c= 16 Solution follows…

Lesson Using the Pythagorean Theorem 3. 3. 2 Independent Practice Calculate the value of a in Exercises 6– 10. 6. 7. am 5 feet 4 feet 8. a cm 7. 5 m a=6 4 cm a = 0. 9 4. 1 cm 4. 5 m a feet a=3 9. 10. units 3 units a=6 a in 3 in a= in 17 Solution follows…

Lesson Using the Pythagorean Theorem 3. 3. 2 Independent Practice 11. Find the length of line AB. 12. Find the perimeter of quadrilateral ABCD y 4 B A 3 A B 2 y 2 1 1 0 3 0 1 2 3 4 5 x 0 – 2 – 1 0 – 1 » 5. 1 units – 2 2 x 1 2 C D + 10 » 12. 8 units 18 Solution follows…

Lesson 3. 3. 2 Using the Pythagorean Theorem Round Up The Pythagorean theorem is really useful for finding missing side lengths of right triangles. If you know the lengths of both legs of a triangle, you can use the formula to work out the length of the hypotenuse. And if you know the lengths of the hypotenuse and one of the legs, you can rearrange the formula and use it to work out the length of the other leg. 19