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Pythagorean theorem
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a, b are legs. c is the hypotenuse (across from the right angle).
Pythagorean Theorem The sum of the squares of the sides of a right triangle is equal to the square of the hypotentuse. We can represent this relationship using a formulaβ¦ π 2 + π 2 = π 2 a, b are legs. c is the hypotenuse (across from the right angle).
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Proving Pythagorean Theorem
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Ok so what was the point? The point of using Pythagorean theorem is to find missing sides of Right Triangles. We can identify the parts of a right triangle, and use them in Pythagorean Theorem!
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Using a2 + b2 = c2 x 15 20 Looking for length of the hypotenuse
π2 + π2 = π2 Pythagorean Theorem Looking for length of the hypotenuse = π2 Substitution = π2 Simplify x 15 625 = π2 Simplify 625 = π 2 20 Square root both sides 25=π Simplify
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Using a2 + b2 = c2 10 6 x Looking for length of a Leg! π2 + π2 = π2
Pythagorean Theorem Looking for length of a Leg! 62 +π2 =102 Substitution 36 + π 2 =100 Simplify 10 6 π 2 =64 Subtraction π 2 = 64 x Square root both sides π=8 Simplify
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You Try! 3 13 8 x x 5 a2 + b2 = c2 a2 + b2 = c2 82 + 32 = c2
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Converse of Pythagorean theorem
If π2 + π2 = π2, then the triangle is a right triangle If π2 + π2> π2 then the triangle is ACUTE If π2 + π2< π2 then the triangle is OBTUSE In english: If we know that the leg squared, plus the other leg squared, is equal to the hypotenuse squaredβ¦Then the triangle must be a RIGHT TRIANGLE!
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Example: Are these the side lengths of a right triangle? π2 + π2 = π2 Pythagorean Theorem 3 =52 Substitution 9 +16 =25 Simplify 4 25 =25 5 Simplify YES! This is a right triangle ο
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You Try!: 9 9 7 6 16 7 Are these the side lengths of a right triangle?
NO! Not a triangle! 16 7 NO! This is a not a right triangle! It is ACUTE!!
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Pythagorean Triples Pythagorean triples are three positive integers such that π 2 + π 2 = π 2 . There are a few that are easy to remember, like the following: 3,4,5 5,12,13 7,24,25 To determine if three sets of numbers form a triple, we plug them into π 2 + π 2 = π 2 and see if we reach equality. For example: = 5 2 9+16=25 25=25
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