Pythagorean Theorem 5.4. Learn the Pythagorean Theorem. Define Pythagorean triple. Learn the...

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Transcript of Pythagorean Theorem 5.4. Learn the Pythagorean Theorem. Define Pythagorean triple. Learn the...

  • 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.a2 + b2 = c2

  • Find the value of x. Give your answer in simplest radical form.a2 + b2 = c2Pythagorean Theorem22 + 62 = x2Substitute 2 for a, 6 for b, and x for c.40 = x2Simplify.Find the positive square root.Simplify the radical.

  • Find the value of x. Give your answer in simplest radical form.a2 + b2 = c2Pythagorean Theorem42 + 82 = x2Substitute 4 for a, 8 for b, and x for c.80 = x2Simplify.Find the positive square root.Simplify the radical.

  • A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 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 a2 + b2 = c2, so they form a Pythagorean triple.a2 + b2 = c2Pythagorean Theorem142 + 482 = c2Substitute 14 for a and 48 for b.2500 = c2Multiply and add.50 = cFind the positive square root.

  • Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.a2 + b2 = c2Pythagorean Theorem82 + 102 = c2164 = c2

  • 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 a2 + b2 = c2, so they form a Pythagorean triple.a2 + b2 = c2Pythagorean Theorem242 + b2 = 262Substitute 24 for a and 26 for c.b2 = 100Multiply and subtract.b = 10Find 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.

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

  • Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.5, 7, 10Since 2< 10 < 12; 5, 7, and 10 can be the side lengths of a triangle.Classify the triangle.Since c2 > a2 + b2, the triangle is obtuse.Add and compare.100 > 74Multiply.Compare c2 to a2 + b2.Substitute the longest side for c.

  • 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

  • Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.7, 12, 16Since 5 < 16 < 19; 7, 12, and 16 can be the side lengths of a triangle.Classify the triangle.Since c2 > a2 + b2, the triangle is obtuse.Add and compare.256 > 193Multiply.Compare c2 to a2 + b2.Substitute the longest side for c.

  • 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

  • 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, 5.2Since .3< 5.2 < 7.9; 3.8, 4.1, and 5.2 can be the side lengths of a triangle.Classify the triangle.Since c2 < a2 + b2, the triangle is acute.Add and compare.27.04 < 31.25Multiply.Compare c2 to a2 + b2.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 Pythagoreans 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 < 9AcuteYes a triangle can be formed 16 < 26 < 36Rightc.Yes a triangle can be formed 5 < 25 < 35Right

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

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

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

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

  • Assignment

    Section 5.4 8 - 30

    *