How to Solve Pythagorean Theorem Word Problems

The Pythagorean Theorem is an equation attributed to the ancient Greek mathematician, Pythagoras. The theorem states that in any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The formula can be written as \(a^2+b^2=c^2\).

An essential part of understanding the Pythagorean Theorem involves recognizing Pythagorean triplets. These are sets of three integers that satisfy the Pythagorean Theorem. Notable examples are \((3,4,5), (5,12,13)\), and \((8,15,17)\). Recognizing these sets can simplify solving Pythagorean Theorem word problems.

A Step-by-step Guide to Solving Pythagorean Theorem Word Problems

Now that we’ve laid the groundwork, let’s delve into the process of solving word problems involving the Pythagorean Theorem.

Step 1: Analyzing the Problem

The first step in solving any word problem is understanding what the problem is asking. Translate the problem into an understandable format and identify the elements related to the Pythagorean Theorem.

Step 2: Sketching the Problem

Visualizing the problem can be extremely helpful. Sketch the triangle described in the problem and label the sides according to the given information. The right angle is usually denoted by a small square, and the hypotenuse, by the letter c.

Step 3: Applying the Pythagorean Theorem

Apply the Pythagorean Theorem to the problem. Plug in the values of the known sides into the theorem and solve for the unknown side.

Step 4: Verifying the Solution

After you’ve found the value of the unknown side, verify your solution by substituting the values back into the theorem. If the sides satisfy the theorem, you’ve correctly solved the problem.

Example of a Pythagorean Theorem Word Problem

Consider a problem where you’re given a right-angled triangle with one side measuring \(5\) units and the hypotenuse measuring \(13\) units. You’re asked to find the length of the other side.

You can use the Pythagorean Theorem to solve this problem. By substituting \(a=5\) and \(c=13\) into the theorem, you can solve for \(b\).

Calculating this will give you \(b^2=13^2-5^2=144\). Therefore, \(b=\sqrt{144}=12\) units. Your verification will involve substituting \(a=5, b=12\), and \(c=13\) into the theorem. The equation \(5^2+12^2=13^2\) checks out, confirming the solution.

The Pythagorean Theorem is a potent tool for solving geometrical problems involving right-angled triangles. With the steps outlined in this guide, you can confidently tackle any word problem that comes your way. Practice is key to mastery, so take time to solve different problems and apply the theorem in real-world situations.