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.
Do this: Read the concept below, then try the quiz or activity.
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.