1. Copyright © Ed2Net Learning, Inc.1 Pythagorean Theorem Grade 7 Pre-Algebra

2. Copyright © Ed2Net Learning, Inc.2 Good Afternoon! Today we will be learning about Pythagorean Theorem Let’s warm up : Find each square root. 1) √64 2) √441 Solve the equation. Round decimal answer to the nearest tenth. 3) a 2 = 300 4) b 2 = 40,000

3. Copyright © Ed2Net Learning, Inc. 3 In words: The square root of a number is one of its two equal factors. In Symbols: If x 2 = y, then x is a square root of y. Definition of square root Let’s review what we did in the last session. Square Roots and Irrational Numbers For example: The square root of 49 is 7 since 7.7 or 7 2 is 49. It is also true that -7.(-7) = 49. So -7 is another square root of 49.

4. Copyright © Ed2Net Learning, Inc. 4 The Square Root Symbol The symbol √, called the radical sign is used to indicate a non-negative square root. √49 indicates the non-negative square root of 49. √49 = 7 -√49 indicates the negative square root of 49. -√49 = -7 Review

5. Copyright © Ed2Net Learning, Inc. 5 a.) √ 25 The symbol √ 25 represents the nonnegative square root of 25. Since 5*5 = 25, √ 25 = 5 Find each square root. b.) √ 100 The symbol √ 100 represents the nonnegative square root of 81. Since 10* 10 = 10, √ 100 = 10 Review

6. Copyright © Ed2Net Learning, Inc. 6 The area of a square is 100 square inches. Find its perimeter. Area = 100 in 2 Answer : Perimeter is 40 inches First find the length of each side. A = s 2 100 = s 2 Replace A with 100. Both 10 and -10, when multiplied by themselves are 100. However, since length can not be negative, s must be 10. Hence, the length of each side is 10inches. Now find the perimeter. P = 4. s P = 4. 10 P = 40 Replace s with 10. Review

7. Copyright © Ed2Net Learning, Inc. 7 List some squares of numbers that are close to 125. 10 2 = 100 11 2 = 121 12 2 = 144 Numbers like 25, 49 and 64 are called perfect squares because when you take the square root then you get an answer that is whole number. What if the number is not perfect square? How do you find the square root of a number like 125? The number 125 is not a perfect square, that is, 125 has no square root hat is a whole number. Continued on the next slide. Review

8. Copyright © Ed2Net Learning, Inc. 8 Since 125 is close to 121 than to 144, the best whole number estimate for √125 would be 11. √ 121 < √125 < √144 √ 11 2 < √125 < √ 12 2 11 < √125 < 12 We know that 125 is greater than 121 or 11 2 and less than 144 or 12 2. So, the square root of 125 should be greater than 11 and less than 12. Review

9. Copyright © Ed2Net Learning, Inc. 9 Irrational Numbers Irrational Numbers: An irrational number is any real number that can not be expressed as a, b where a and b are integers and b is not equal to 0. Informally, this means numbers that cannot be represented as simple fractions. It can be deduced that they also cannot be represented as terminating or repeating decimals, Determine whether the number is rational or irrational. 0.22222... The three dots means that the 2s keep repeating. This is repeating decimal, so it can be expressed as a fraction. 0.22222... = 2 9 Thus, it is a rational number. Review

10. Copyright © Ed2Net Learning, Inc. 10 Rational number are numbers that can be expressed as a ratio of two integers, where the divisor are not zero. Irrational number are numbers that can be named by non terminating, non repeating decimals. Real numbers Rational numbers Integers Whole numbers Irrational numbers The sets of real numbers includes both the rational numbers and irrational numbers. Review

11. Copyright © Ed2Net Learning, Inc. 11 Solve the equation. Round decimal answer to the nearest tenth. y 2 = 75 Solution: y 2 = 75 y = √ 75 or - √75 Take the square root of each side. 75 = 8.660254038 y ≈ 8.7 or y ≈ -8.7

12. Copyright © Ed2Net Learning, Inc. 12 Today we will discuss about Pythagorean Theorem The Pythagorean Theorem shows how the legs and hypotenuse of a right triangle are related. legs hypotenuse In a right triangle, the two shortest sides are legs. The longest side, which is opposite the right angle, is the hypotenuse.

13. Copyright © Ed2Net Learning, Inc. 13 In words: In a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs. In Symbols: a 2 +b 2 = c 2. Pythagorean Theorem a b c If we know the lengths of two sides of a right angled triangle, then Pythagoras' Theorem allows us to find the length of the third side.

14. Copyright © Ed2Net Learning, Inc. 14 The lengths of the sides of a right triangle are 5 ft and 12 ft. Find the length of the hypotenuse. 1. Use the Pythagorean Theorem c 2 = a 2 +b 2 2. Replace a with 5 and b with 12 c 2 = 5 2 +12 2 = 25 +144 = 169 c= √ 169 = 13 The length of the hypotenuse = 13 ft.

15. Copyright © Ed2Net Learning, Inc. 15 You can use algebra to find any missing value, as in the following examples. a 2 + b 2 = c 2 9 2 + b 2 = 15 2 81 + b 2 = 225 Take 81 from both sides b 2 = 144 b = √144 b = 12 9 15 b

16. Copyright © Ed2Net Learning, Inc. 16 The converse of Pythagorean Theorem allows you to substitute the lengths of the sides of a triangle into the equation. c 2 = a 2 +b 2 To check whether a triangle is a right triangle, if the Pythagorean equation is true the triangle is a right triangle.

17. Copyright © Ed2Net Learning, Inc. 17 Is a triangle with sides 12 m, 15 m, and 20 m a right triangle? a 2 +b 2 = c 2 Write the equation for Pythagorean Theorem 12 2 +15 2 = 20 2 Replace a and b with the shorter lengths and c with the longest length 144+225 = 400 Simplify 369 ‡ 400 The triangle is not a right triangle.

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20. Copyright © Ed2Net Learning, Inc. 20 Assignments The lengths of the sides of a right triangle are given. Find the length of the hypotenuse. 1) a = 12, b = 6 2) a = 2, b = 5 3) a = 7, b = 7 4) a = 18, b = 5

21. Copyright © Ed2Net Learning, Inc. 21 In a right angled triangle, if a and b are the measures of the legs and c is the measure of the hypotenuse, find each missing measure. Round decimal answers to the nearest tenths. 5) a = 4, b = 5 6) a = 3, c = 7 7) b = 12, c = 35 8) a = 15, b = 16.7

22. Copyright © Ed2Net Learning, Inc. 22 The measurement of three sides of a triangle are given. Determine whether each triangle is a right triangle. 9) 9 cm, 12 cm, 15cm 10) 9 ft, 12 ft, 15 ft

23. Copyright © Ed2Net Learning, Inc. 23 Pythagorean Theorem The Pythagorean Theorem shows how the legs and hypotenuse of a right triangle are related. legs hypotenuse In a right triangle, the two shortest sides are legs. The longest side, which is opposite the right angle, is the hypotenuse. Very Good! Let's Review

24. Copyright © Ed2Net Learning, Inc. 24 In words: In a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs. In Symbols: a 2 +b 2 = c 2. Pythagorean Theorem a b c If we know the lengths of two sides of a right angled triangle, then Pythagoras' Theorem allows us to find the length of the third side. Review

25. Copyright © Ed2Net Learning, Inc. 25 The lengths of the sides of a right triangle are 5 ft and 12 ft. Find the length of the hypotenuse. 1. Use the Pythagorean Theorem c 2 = a 2 +b 2 2. Replace a with 5 and b with 12 c 2 = 5 2 +12 2 = 25 +144 = 169 c= √ 169 = 13 The length of the hypotenuse = 13 ft. Review

26. Copyright © Ed2Net Learning, Inc. 26 You can use algebra to find any missing value, as in the following examples. a 2 + b 2 = c 2 9 2 + b 2 = 15 2 81 + b 2 = 225 Take 81 from both sides b 2 = 144 b = √144 b = 12 9 15 b Review

27. Copyright © Ed2Net Learning, Inc. 27 The converse of Pythagorean Theorem allows you to substitute the lengths of the sides of a triangle into the equation. c 2 = a 2 +b 2 To check whether a triangle is a right triangle, if the Pythagorean equation is true the triangle is a right triangle. Review

28. Copyright © Ed2Net Learning, Inc. 28 Is a triangle with sides 12 m, 15 m, and 20 m a right triangle? a 2 +b 2 = c 2 Write the equation for Pythagorean Theorem 12 2 +15 2 = 20 2 Replace a and b with the shorter lengths and c with the longest length 144+225 = 400 Simplify 369 ‡ 400 The triangle is not a right triangle. Review

29. Copyright © Ed2Net Learning, Inc. 29 You have done a nice job. See you in the next session.