GED Mathematical Reasoning: Pythagorean Theorem
In this chapter, we will be discussing the Pythagorean Theorem – one of the most famous theorems in mathematics. A “theorem” is simply a statement of truth.
The Pythagorean Theorem states the very special relationship between the sides of a right triangle. It is often used in architecture and construction. Recall that a right triangle contains a right angle, measuring 90 degrees.
The two sides that make up the 90 degree angle are called “legs” and the side directly across from the right angle, which is the longest of the three sides, is called the “hypotenuse.”
It is common to label the legs with the letters a and b and the hypotenuse with the letter c.
The Pythagorean Theorem states that given ANY right triangle, the following relationship holds true:
In equation form, the Pythagorean Theorem is:
Where a and b are the lengths of the legs and c is the hypotenuse of a right triangle.
If we’re given two side measures of a right triangle, we can use the Pythagorean theorem to calculate the length of the third side, essentially using the Pythagorean Theorem as a formula.
Calculate the length of the missing side.
In example one, we have a right triangle and the lengths of both legs are given. One is 5 inches and the other is 13 inches. We are being asked to find the length of the missing side — the hypotenuse – so this is a perfect situation for applying the Pythagorean Theorem.
Let’s start by labeling the triangle – this will help us when we substitute numbers for variables in the formula. It is a good rule of thumb to always label the legs with the letters a and b and the hypotenuse with c.
Applying the Pythagorean Theorem, we’ll substitute 5 for a and 13 for b.
To find the length of the hypotenuse, c, we’ll solve this equation for c.
To solve for c and get it by itself on one side of the equal sign, we need to “get rid” of the square power. The operation that will un-do a square power is the square root.
Since the number 194 is not a perfect square, we will use the calculator to find its value, which is approximately 13.9
So the hypotenuse of this right triangle measures approximately 13.9 inches.
As a side note, you will learn more about solving equations by squaring both sides as you continue your studies. Specifically you will learn that when squaring both sides of an equation, the positive AND negative root must be accounted for.
To complete a home project, Sandra places a 12 foot ladder 6 feet from the base of her house. How far up the wall will the ladder reach?
Let’s first create a visual graphic depicting the situation.
The 12 foot ladder is leaning against the side of the house and the base of the ladder is resting on the ground, 6 feet from the base of the house. Notice that a right triangle is formed whereby the ground and side of the house are the legs and the ladder is the hypotenuse. We’re given that one leg measures 6 feet and that the hypotenuse measures 12 feet. Based on this information we can use the Pythagorean Theorem to find the length of the missing leg.
Again, let’s start by labeling the triangle. We’ll label the legs a and b and the hypotenuse as c.
Applying the Pythagorean Theorem, we’ll substitute 6 for b and 12 for c.
To find the length of the missing leg, a, we’ll solve this equation for a
From here, we will square root both sides of the equation to get a by itself.
Since 108 is not a perfect square, let’s use the calculator to find the value of the square root of 108. It is approximately 10.4
So the missing side length is approximately 10.4.
In closing, as you have seen it is very helpful to label the sides of the triangle before applying the Pythagorean Theorem using the letters a and b for the legs and the letter c for the hypotenuse.
It is very important to always use the letter c for the hypotenuse – the side directly across from the right angle. However, when it comes to the legs, it doesn’t really matter which leg we label as a and which we label as b. The reason for this goes back to a property of addition which says that the order in which we add doesn’t matter. 2 plus 3 and 3 plus 2 has the same result the way that a squared plus b squared is the same as b squared plus a squared.