Pythagorean theorem.

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The Pythagorean Theorem and its Converse

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Presentation transcript:

Pythagorean theorem

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 formula… 𝑎 2 + 𝑏 2 = 𝑐 2 a, b are legs. c is the hypotenuse (across from the right angle).

Proving Pythagorean Theorem https://www.youtube.com/watch?v=BNCj-K2hd_k https://www.youtube.com/watch?v=vbG_YBTiN38

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!

Using a2 + b2 = c2 x 15 20 Looking for length of the hypotenuse 𝑎2 + 𝑏2 = 𝑐2 Pythagorean Theorem Looking for length of the hypotenuse 152 + 202 = 𝑐2 Substitution 225 + 400 = 𝑐2 Simplify x 15 625 = 𝑐2 Simplify 625 = 𝑐 2 20 Square root both sides 25=𝑐 Simplify

Using a2 + b2 = c2 10 6 x Looking for length of a Leg! 𝑎2 + 𝑏2 = 𝑐2 Pythagorean Theorem Looking for length of a Leg! 62 +𝑏2 =102 Substitution 36 + 𝑏 2 =100 Simplify 10 6 𝑏 2 =64 Subtraction 𝑏 2 = 64 x Square root both sides 𝑏=8 Simplify

You Try! 3 13 8 x x 5 a2 + b2 = c2 a2 + b2 = c2 82 + 32 = c2

Converse of Pythagorean theorem If 𝑎2 + 𝑏2 = 𝑐2, then the triangle is a right triangle If 𝑎2 + 𝑏2> 𝑐2 then the triangle is ACUTE If 𝑎2 + 𝑏2< 𝑐2 then the triangle is OBTUSE In english: If we know that the leg squared, plus the other leg squared, is equal to the hypotenuse squared…Then the triangle must be a RIGHT TRIANGLE!

Example: Are these the side lengths of a right triangle? 𝑎2 + 𝑏2 = 𝑐2 Pythagorean Theorem 3 32 +42 =52 Substitution 9 +16 =25 Simplify 4 25 =25 5 Simplify YES! This is a right triangle 

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!!

Pythagorean Triples Pythagorean triples are three positive integers such that 𝑎 2 + 𝑏 2 = 𝑐 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 𝑎 2 + 𝑏 2 = 𝑐 2 and see if we reach equality. For example: 3 2 + 4 2 = 5 2 9+16=25 25=25