5-3A The Pythagorean Theorem You used the Pythagorean Theorem to develop the Distance Formula. Use the Pythagorean Theorem. Use the Converse of the Pythagorean Theorem.

Pythagorean Theorem The Pythagorean Theorem is used to calculate the length of any side of a right triangle when the lengths of the other two sides are known. Which side is the hypotenuse? Which sides are the legs? a b c

Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. a2 + b2 = c2 a c b p. 547

p. 547

Find the length of the third side of the right triangle ∆STU. ST=3, TU = 4 ST2 + TU2 = SU2 32 + 42 = SU2 9 + 16 = SU2 25 = SU2 5 = SU S T U

The side opposite the right angle is the hypotenuse, so c = x. A. Find x. The side opposite the right angle is the hypotenuse, so c = x. a2 + b2 = c2 Pythagorean Theorem 42 + 72 = c2 a = 4 and b = 7 65 = c2 Simplify. Take the positive square root of each side. Answer:

a2 + b2 = c2 Pythagorean Theorem x2 + 82 = 122 b = 8 and c = 12 B. Find x. The hypotenuse is 12, so c = 12. a2 + b2 = c2 Pythagorean Theorem x2 + 82 = 122 b = 8 and c = 12 x2 + 64 = 144 Simplify. x2 = 80 Subtract 64 from each side. Take the positive square root of each side and simplify. Answer:

Find the length of the third side of the right triangle ∆STU. ST=7, SU = 10 ST2 + TU2 = SU2 72 + TU2 = 102 49 + TU2 = 102 49 + TU2 = 100 TU2 = 51 S T U

A. Find x. A. B. C. D.

B. Find x. A. B. C. D.

Pythagorean Triple A Pythagorean triple is 3, 4, and 5. 32 + 42 = 52 Pythagorean Triples are special sets of numbers that all the numbers are positive integers. A Pythagorean triple is 3, 4, and 5. 32 + 42 = 52 9 + 16 = 25

Pythagorean Triples under 100 (3, 4, 5)( 5, 12, 13)( 7, 24, 25)( 8, 15, 17) ( 9, 40, 41)(11, 60, 61)(12, 35, 37) (13, 84, 85)(16, 63, 65)(20, 21, 29) (28, 45, 53)(33, 56, 65)(36, 77, 85) (39, 80, 89)(48, 55, 73)(65, 72, 97) Formula: Suppose that m and n are two positive integers, with m < n. Then n2 - m2, 2mn, and n2 + m2 is a Pythagorean triple.

p. 548

Find the length of the missing side. 8 6 12 13 25 7 5 Triple = 5, 12, 13 24 Triple = 7, 24, 25 10 Triple = 6, 8, 10

Use a Pythagorean triple to find x. Explain your reasoning. Notice that 24 and 26 are multiples of 2: 24 = 2 ● 12 and 26 = 2 ● 13. Since 5, 12, 13 is a Pythagorean triple, the missing leg length x is 2 ● 5 or 10. Answer: x = 10 Check: 242 + 102 = 262 Pythagorean Theorem ? 676 = 676 Simplify. 

Use a Pythagorean triple to find x. B. 15 C. 18 D. 24

If you have the lengths of three sides of a triangle, you can use the converse of the Pythagorean Theorem to prove it is a right triangle. p. 550

You can also use side lengths to classify an acute or obtuse triangle. p. 550    

The side lengths 9, 12, and 15 can form a triangle. A. Determine whether 9, 12, and 15 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem. 9 + 12 > 15  9 + 15 > 12  12 + 15 > 9  The side lengths 9, 12, and 15 can form a triangle. Step 2 Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. c2 = a2 + b2 Compare c2 and a2 + b2. ? 152 = 122 + 92 Substitution ? 225 = 225 Simplify and compare. Answer: Since c2 = a2 + b2, the triangle is a right triangle.

The side lengths 10, 11, and 13 can form a triangle. B. Determine whether 10, 11, and 13 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem. 10 + 11 > 13  10 + 13 > 11  11 + 13 > 10  The side lengths 10, 11, and 13 can form a triangle. Step 2 Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides. c2 = a2 + b2 Compare c2 and a2 + b2. ? 132 = 112 + 102 Substitution ? 169 < 221 Simplify and compare. Answer: Since c2 < a2 + b2, the triangle is acute.

A. Determine whether the set of numbers 7, 8, and 14 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. A. yes, acute B. yes, obtuse C. yes, right D. not a triangle

What is the Pythagorean Theorem? a2 + b2 = c2 Why is it important? It is used to calculate the length of any side of a right triangle when the lengths of the other two sides are known. What is a Pythagorean Triple? Pythagorean Triples are special sets of numbers that all the numbers are positive integers.

8-2 Assignment Page 552, 8-28 even