Math 8 ( Period 1)

The Pythagorean Theorem 5-5

This theorem relates the legs of a right triangle to its hypotenuse.
The legs form the right angle.
The legs are called  a and b. It does not matter which is a and which one you designate as b  because you will add them and adding is COMMUTATIVE)

The hypotenuse is across from the right angle and it is called   c.
You can find the third side of a right triangle as long as you know the other two sides

The Pythagorean Theorem formula: a² + b² = c²
The sum of each leg squared is equal to the sum of the hypotenuse squared.

 A couple of important relationships:
The hypotenuse is always the LONGEST individual SIDE
The sum of the two legs is always GREATER than the hypotenuse (that’s why I call the hypotenuse the  “ shortcut home”)  

After squaring the two sides that you know you will need to find the square root of that number to find the length of the missing side.

EASIEST—is finding the hypotenuse (c)
You have a right triangle with sides 8cm  and 15cm plugging this into the Pythagorean Theorem
a² + b² = c²
8² + 15² = c²
64 + 225 = c²
289 = c²            If you know your perfect squares ( HINT) you know
17 = c
That means the length of the hypotenuse is 17 cm

Note: Sometimes the solution is not a whole number. You might need to use a calculator to find the square root of a number.

A LITTLE HARDER_ finding  a missing leg ( either a or b)
This time you are given that a  leg is 5 feet and the hypotenuse is 13 ft
a² + b² = c²
5² + b² = 13²   Make sure to put those two given numbers in the formula correctly
25 + b² = 169
b² = 169-25
b²= 144     You know
b = 12

The Converse of the Pythagorean Theorem:
If you add the squares of the legs and that sum EQUALS the square of the longest side—then it is a RIGHT TRIANGLE.
if you know the sides of a triangle without drawing it, you can plug them into the formula and determine whether it is a right triangle!
If the formula works then the triangle is a right triangle.

Note:
If you add the squares of the two smaller sides and that sum is GREATER THAN the square of the longest side, you have an ACUTE TRIANGLE

If you add the squares of the two smaller sides and that sum is LESS THAN the square of the longest side,  you have an OBTUSE TRIANGLE