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The Pythagorean Theorem

The Pythagorean Theorem. Given a right triangle with legs of lengths a and b and, a hypotenuse of length c . The hypotenuse will always be opposite of the right angle. It will also always be the longest of the three sides of the triangle. The Pythagorean Theorem.

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The Pythagorean Theorem

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  1. The Pythagorean Theorem

  2. Given a right triangle with legs of lengths a and b and, a hypotenuse of length c. The hypotenuse will always be opposite of the right angle. It will also always be the longest of the three sides of the triangle. The Pythagorean Theorem

  3. Draw a square using each side of the right triangle (remember this is extending each side of the triangle outward to create the square). Let’s Take a Closer Look!

  4. Now: Knowing that the area of a square is: A = s ∙ s = s2; then a ∙ a = a2; b ∙ b = b2, and c ∙ c = c2. • Now using: a2 + b2 = c2 leg2+ leg2= hypotenuse2 • Last: Don’t forget that in the end you actually want a, b, or c; not its square. So, we must find the square root of whatever side we are looking for. Let’s Take a Closer Look!

  5. Based on what we just learned, we can define the Pythagorean Theorem as the sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse. So What is The Pythagorean Theorem?

  6. Leg = 12; Leg = 16; find the hypotenuse a = 12; b = 16; c = ? 122 + 162 = c2 144 + 256 = c2 400 = c2 √(400) = √(c2) Remember we want c! 20 = c Try It!

  7. Leg = ?; Leg = 18; hypotenuse = 20 (we are looking for a leg) a2 + 182 = 202 a2 + 324 = 400 -324 = -324 a2 = 76 √(a2) = √(76) a ≈ 8.7 Try It!

  8. Leg = 3; Leg = 7; hypotenuse = ? a2 + b2 = c2 32 + 72 = c2 9 + 49 = c2 58 = c2 √(58) = √(c2) 7.6 ≈ c Try It!

  9. Leg = 10; Leg = ?; hypotenuse = 14 a2 + b2 = c2 102 + b2 = 142 100 + b2 = 196 -100 = -100 b2 = 96 √(b2) = √(96) b ≈ 9.8 Try It!

  10. Now, think back – This formula a2 + b2 = c2 only works if the triangle is a right triangle. So, if we are given 3 lengths of a triangle we can apply the formula a2 + b2 = c2 to determine if a triangle with the given lengths is a right triangle or not, because a2 + b2 must equal c2. • Now let’s do it! 42 + 52 = 62 16 + 25 = 36 41 ≠ 36

  11. Notice: Look at any right triangle. What side is always the longest? • Side c; the hypotenuse or the side opposite the right angle. • So; when you are given 3 lengths of a triangle, which one must always be “c”? The longest length one! The Hypotenuse

  12. We are given the three sides of our triangle: 6, √(13), and √(23). Which one of these must be “c”? 6 must be “c” because it has the longest length. Now we are ready to work the problem . (√(13))2 + (√(23))2 = 62 13 + 23 = 36 36 = 36 Yes, it is a right triangle! Let’s look at another example:

  13. The three sides of our triangle are: 8, 10, and 12 82+ 102 = 122 64 + 100 = 144 164 ≠ 144 No, it is NOT a right triangle! • If we are given: 2, 7, and 11 22 + 72 = 112 4 + 49 = 121 53 ≠ 121 No, it is NOT a right triangle! Just two more!

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