Solving Equations with the Pythagorean Theorem in Geometry: Help Your Child Learn How

The Pythagorean Theorem is one of the most famous theorems of introductory geometry and it is often encountered well before high school geometry, in middle school classes. It is named after an ancient Greek mathematician, Pythagoras, who is credited with having discovering it.

Prerequisite Skills:
Your child will need to have solid background skills in prealgebra in order to work with the Pythagorean Theorem; including knowing squares, square roots, and the technique of solving one equation with one variable.

If your high school geometry student can’t solve the sample equations that I’ve provided (see below) by him or herself, I would recommend getting him or her some geometry help by practicing working with the perfect squares at least up to 225.

Failure to recognize perfect squares makes working with the Theorem much more confusing than it needs to be.

What is the Pythagorean Theorem:
The theorem relates the lengths of the three sides of a right triangle, stating that the sum of the squares of the two shorter sides, called the legs of the triangle, is equal to the square of the longer side, the side opposite the right angle, called the hypotenuse.

3²+4²=X²
9+16=X²
25=X²
So, X=5

The second variety is when the unknown length is one of the two legs. For example: one leg is 12 and the hypotenuse is 13.

Then the equation will be:
X²+12²=13²
X²+144=169
X²=25
So X=5

When you write these equations, you just need to remember to put the length of the longest side by itself.

Real Life Applications of the Theorem
There are many applications of the Pythagorean Theorem, and that’s why it plays a prominent role in Geometry problems. There are many professions that use the Theorem, such as architects and ship navigators.

For example: if a ship sails West for 7 miles and North for 24 miles, how far is it from it’s starting point?

To solve this a navigator could make a sketch which will show that the ship has sailed two legs of a right triangle, and the unknown distance is the hypotenuse.

So the equation is:
7² + 24²=X²
49+576=X²
625=X²
25=X

By the way, there are a host of practical sreasons every child needs a high school math education, here is an article dedicated to this discussion.

Common Mistakes:
The most common mistake is to set up the equation with one of the legs by itself. You just need to remember that it’s always the longest side that’s by itself in the equation.

Also, many math books complicate the situation by writing the Pythagorean Theorem in a second form with subtraction when you are using it to find one of the legs. This isn’t necessary and it can make a student think there is another equation, when its really just a second form of the basic equation.

That’s why I recommend always using the one form, A²+B²=C².

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