The Pythagorean Theorem Converse & Triangle Inequality Theorem

1. The Pythagorean Theorem Converse & Triangle Inequality Theorem • Pythagoras, circa 570 BC.

2. Pythagorean Theorem Review • In a right triangle, the sum of the squares of each leg equals the square of the hypotenuse.

3. Pythagorean Theorem Review • Therefore, in this triangle, the legs are a and b. The hypotenuse is c. So, the pythagorean theorem states: a2 + b2 =c2

4. Pythagorean Theorem CONVERSE • The Pythagorean Theorem ONLY works for right triangles. Therefore, the converse of the Pythagorean Theorem is true. The converse reads…if a2 + b2 =c2, where c is the longest side of a triangle, then the triangle is a right triangle.

5. Pythagorean Theorem CONVERSE • So, if you don’t know if the triangle is a right triangle, but the Pythagorean Theorem “works”, then it IS a right triangle. • Example: If a = 3, b = 4, and c = 5, is the triangle a right triangle? Work out the Pythagorean theorem on your paper.

6. Pythagorean Theorem CONVERSE • a2 + b2 ? C2 • 32 + 42 ? 52 • 9 + 16 ? 25 • 25 = 25 …..So, the triangle IS a right triangle!

7. Pythagorean Theorem CONVERSE • A triangle has side lengths of 6, 7, and 10. Is the triangle a right triangle?

8. REMEMBER! The Pythagorean Theorem ONLY works for RIGHT triangles!

9. The Triangle Inequality Theorem • IN ANY triangle, the sum of any 2 sides MUST be larger than the third side! • If this doesn’t occur, you will not be able to create ANY kind of triangle. • Example: You have wood that is 3cm, 5cm, and 9 cm long. Can you create a triangle with these 3 pieces of wood?

10. The Triangle Inequality Theorem • NO. You could NOT create a triangle. • 3 + 5 = 8. 8 is not bigger than 9, so you would have 2 pieces that would not connect!

11. REVIEW REVIEW REVIEW REVIEW • The PYTHAGOREAN THEOREM ONLY WORKS FOR RIGHT TRIANGLES. • THE SUM OF ANY 2 SIDES OF A TRIANGLE MUST BE LARGER THAN THE THIRD SIDE.