WHAT IS THE PYTHAGOREAN THEOREM AND HOW DO WE USE IT TO ANALYZE RIGHT TRIANGLES? AGENDA: Simplifying radicals warmup Pythagorean theorem notes and practice

The legs are always the two sides that form the right angle and are shorter than the hypotenuse. Definition of a Right Triangle – a triangle with exactly one right angle. legs The hypotenuse is always across from the right angle and is always the longest side. This is a review slide. You do not need to write anything down.

Pythagorean Theorem – In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. a b c a 2 + b 2 = c 2

Example 1Example c b

Example 3Example c 9 18 b

Most of the time, at least one of your side lengths in a right triangle will be an irrational number. A Pythagorean Triple consists of three INTEGERS that fulfill the Pythagorean Theorem. abc a 2 + b 2 = c 2

1. A right triangle has leg lengths 5 and 5, find the length of the hypotenuse. 5 c 2. A triangle has side lengths 3, 3, and 3. Is the triangle a right triangle? Show work. If the triangle was a right triangle, the longest side would be the hypotenuse. In this case, 3. Check to see if the Pythagorean Theorem works = = 18 Yes it is a right triangle.

3. A rectangle has side lengths 5in and 12in. Find the length of each diagonal. (Hint: it may help to draw a diagram.) 5 12 c = c = c = c 2 13 = c The diagonals of a rectangle are congruent, therefore each diagonal would have the same length: 13 inches.

4. The perimeter of a square is 40cm. Find the length of each diagonal. (Hint: find the length of each side of the square first). 10 c = c = c 2 The length of each diagonal is or cm. The side length of the square is 10 cm.

Converse of the Pythagorean Theorem: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. c 2 = a 2 + b 2 a b c

Is the triangle with the given lengths a right triangle? 4, 5, = 6 2 ? 41 = 36 ? No! 10, 24, 264, 4, 9 Yes! This triangle cannot be constructed!! < = 26 2 ? = 676 ? Yes!

Theorem 2: If c 2 < a 2 + b 2, then m  C < 90 and  ABC is an acute triangle. A B C c b a  C is always the largest angle; c is always the longest side.

Theorem 3: If c 2 > a 2 + b 2, then m  C > 90 and  ABC is an obtuse triangle. A B C c ba  C is always the largest angle; c is always the longest side.

Use the side lengths of the triangle to determine whether it is acute, right or obtuse. If c 2 = a 2 + b 2  right triangle. If c 2 < a 2 + b 2  acute triangle. If c 2 > a 2 + b 2  obtuse triangle. Example 1Example 2Example 3 29, 20, 2120, 21, 3020, 21, ? = 841 RightObtuseAcute 900 ? > ? < 841

Use the side lengths of the triangle to determine whether it is acute, right or obtuse. If c 2 = a 2 + b 2  right triangle. If c 2 < a 2 + b 2  acute triangle. If c 2 > a 2 + b 2  obtuse triangle. Example 4 8,, 16 Right 256 ? = 256 NOTE: If a radical is GIVEN, You may need to convert into a decimal to compare before completing your setup!