Pythagorean TheoremBasic Math
Copyright 2001 Lynda Greene
The Formula
The Pythagorean Theorem is used to find the length of a side of a right triangle.
The sides are labeled a, b and c. The longest side “c” is the side opposite the right angle and is called the hypotenuse. The other two sides are called legs and they are labeled a and b.
The formula is: a2 + b2 = c2
Just plug in the two lengths you are given and solve for the third.
Examples:
The problem is given in one of two ways:
1) You are given the lengths of the two legs and must find the hypotenuse.
2) You are given the lengths of one leg and the hypotenuse and must find the other leg.
Type 1: A right triangle has legs whose lengths are 3 in. and 4 in. How long is the third side.
a2 + b2 = c2
(3)2 + (4)2 = c2
9 + 16 = c2
25 = c2
(take the square root of both sides) Answer 5 = c
Type 2: A right triangle has one side 5 cm long, and the hypotenuse is 13 cm. How long is the third side.
a2 + b2 = c2
(5)2 + b2 = (13)2
25 + b2 = 169
(subtract 25 from both sides)
b2 = 144
(take the square root of both sides)
Answer b = 12
Favorite tricks on tests
Mistakes when solving right-triangle problems don’t usually happen because a student can’t solve the formula. It’s usually caused by the student not recognizing the shape as a right triangle, so s/he doesn’t realize that s/he is supposed to use the Pythagorean Theorem. Here is a list of favorite types of right-triangle problems that tend to show up on tests. I like to call them “right-triangles in disguise”.
Example 1: There is some rectangular object (a park, picture frame, swimming pool, floor of a room), and you are given the lengths of two sides and are asked for the diagonal length from one corner to the other.
Example 2: You are given the length of one side and there is a string of some length stretched from one corner to the other across the figure. This is the one leg, hypotenuse type of right triangle problem. The string is the diagonal side.
In both cases, use the formula. The diagonal side is always the hypotenuse.
Example 3:
They usually won’t write the
numbers on the same triangle.
Example 4: The Ladder,Tree or Flagpole right triangle problem
There is a ladder leaning up against the side of a building. It is 18 ft long and the distance from the base of the ladder to the base of the building is 5 ft. How tall is the building?
There is a tree (flagpole or building) that has a kite stuck at the top. The kite string is stretched from the top of the tree to something on the ground. The string is 18 ft long and the distance from the bottom of the string to the base of the tree is 5 ft. How tall is the tree?
This is the same problem using different objects. The leaning ladder (or the kite string) is the hypotenuse, the height of the building (tree or flagpole) and the distance along the ground from the ladder(wire) to the building are the legs.
This same problem may also be re-worded by giving you the two legs and asking for the hypotenuse.