Pythagorean theorem: word problems
The Pythagorean theorem (a² + b² = c²) is a powerful tool for solving problems involving right triangles. It allows you to find a missing side length, which is incredibly useful for navigation, construction, and geometry.
Concept
The Pythagorean theorem applies to any **right triangle**. It describes the relationship between the lengths of the two shorter sides (legs) and the longest side (hypotenuse). **The Formula: a² + b² = c²** * **a** and **b** are the **legs** (the two sides that form the right angle). * **c** is the **hypotenuse** (the side opposite the right angle; it is always the longest side). **Steps to Solve a Word Problem:** 1. **Draw a diagram.** This is the most important step! Sketch the situation and identify where the right triangle is. 2. **Label the sides.** Identify what you know (a, b, or c) and what you need to find. 3. **Substitute** the known values into the formula a² + b² = c². 4. **Solve** for the unknown side. This may involve squaring numbers, adding, subtracting, and finally, taking a square root. 5. **Check your answer.** Does it make sense in the context of the problem? The hypotenuse must always be the longest side. **Example 1: Finding the Hypotenuse** * **Problem:** You walk 3 miles east and then 4 miles north. How far are you from your starting point in a straight line? 1. **Diagram:** Draw a point, then a line 3 miles east, then a line 4 miles north. The straight-line distance back to the start forms the hypotenuse of a right triangle. 2. **Label:** The legs are a = 3 and b = 4. We need to find the hypotenuse, c. 3. **Substitute:** 3² + 4² = c² 4. **Solve:** * 9 + 16 = c² * 25 = c² * √25 = c * c = 5 5. **Answer:** You are 5 miles from your starting point. **Example 2: Finding a Leg** * **Problem:** A 13-foot ladder is leaning against a wall. The base of the ladder is 5 feet away from the wall. How high up the wall does the ladder reach? 1. **Diagram:** The ladder is the hypotenuse, the wall and ground are the legs. 2. **Label:** The hypotenuse c = 13. One leg (the ground distance) is b = 5. We need to find the other leg (the height on the wall), a. 3. **Substitute:** a² + 5² = 13² 4. **Solve:** * a² + 25 = 169 * Subtract 25 from both sides: a² = 144 * √144 = a * a = 12 5. **Answer:** The ladder reaches 12 feet up the wall. **Key Idea:** Look for right angles in word problems. Words like "north," "south," "east," "west," "up," "down," "height," and "ground" often signal the legs of a right triangle. Diagonals, direct distances, and ramps are often the hypotenuse.
Try it
Practice: Pythagorean theorem: word problems.