Pythagorean Theorem 5.4

Learn the Pythagorean Theorem. Define Pythagorean triple. Learn the Pythagorean Inequality. Solve problems with the Pythagorean Theorem and Inequality.

The Pythagorean Theorem is probably the most famous mathematical relationship. It states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. a 2 + b 2 = c 2

Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 Pythagorean Theorem = x 2 Substitute 2 for a, 6 for b, and x for c. 40 = x 2 Simplify. Find the positive square root. Simplify the radical.

Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 Pythagorean Theorem = x 2 Substitute 4 for a, 8 for b, and x for c. 80 = x 2 Simplify. Find the positive square root. Simplify the radical.

A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2 is called a Pythagorean triple.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. The side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2, so they form a Pythagorean triple. a 2 + b 2 = c 2 Pythagorean Theorem = c 2 Substitute 14 for a and 48 for b = c 2 Multiply and add. 50 = c Find the positive square root.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. a 2 + b 2 = c 2 Pythagorean Theorem = c = c 2 The side lengths do not form a Pythagorean triple because is not a whole number.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. The side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2, so they form a Pythagorean triple. a 2 + b 2 = c 2 Pythagorean Theorem b 2 = 26 2 Substitute 24 for a and 26 for c. b 2 = 100 Multiply and subtract. b = 10 Find the positive square root.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. No. The side length 2.4 is not a whole number.

The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths.

You can also use side lengths to classify a triangle as acute or obtuse. A B C c b a

To understand why the Pythagorean inequalities are true, consider ∆ABC.

The third side of a triangle is less than the sum and greater than the difference of the other two sides. Remember!

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 5, 7, 10 Since 2< 10 < 12; 5, 7, and 10 can be the side lengths of a triangle. Classify the triangle. Since c 2 > a 2 + b 2, the triangle is obtuse. Add and compare. 100 > 74 Multiply. Compare c 2 to a 2 + b 2. Substitute the longest side for c. c 2 = a 2 + b 2 ? 10 2 = ? 100 = ?

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. Determine if the measures form a triangle. 5, 8, 17 Since 3 < 17 < 13, these cannot be the side lengths of a triangle.

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 7, 12, 16 Since 5 < 16 < 19; 7, 12, and 16 can be the side lengths of a triangle. Classify the triangle. Since c 2 > a 2 + b 2, the triangle is obtuse. Add and compare. 256 > 193 Multiply. Compare c 2 to a 2 + b 2. Substitute the longest side for c. c 2 = a 2 + b 2 ? 16 2 = ? 256 = ?

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. Determine if the measures form a triangle. 11, 18, 34 Since 7 < 34 < 29 ; these cannot be the sides of a triangle.

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 3.8, 4.1, 5.2 Since.3< 5.2 < 7.9; 3.8, 4.1, and 5.2 can be the side lengths of a triangle. Classify the triangle. Since c 2 < a 2 + b 2, the triangle is acute. Add and compare < Multiply. Compare c 2 to a 2 + b 2. c 2 = a 2 + b 2 ? = ? = ? Substitute the longest side for c.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 13; yes; the side lengths are nonzero whole numbers that satisfy Pythagorean’s Theorem.

Tell if the measures 7, 11, and 15 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. yes; obtuse

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. a. b. Yes a triangle can be formed 1 < 6 < 9 Acute Yes a triangle can be formed 16 < 26 < 36 Right c. Yes a triangle can be formed 5 < 25 < 35 Right

Find the exact answer of the missing side. a. b. c. d. e. f.

a. b. c. x = 25 x = 34 d. e. f. Find the exact answer of the missing side.

a. b. c. x = 15 x = 51 d. e. f. x = 60 x = 30 Find the exact answer of the missing side.

a. b. c. x = 15 x = 51 d. e. f. Find the exact answer of the missing side. x = 82

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