Overview
This lecture explains how to use the Pythagorean theorem to calculate the length of a missing side in a right triangle, with step-by-step examples.
The Pythagorean Theorem
- The Pythagorean theorem applies only to right triangles.
- The formula is: (leg)^2 + (leg)^2 = (hypotenuse)^2.
- The hypotenuse is always the side opposite the right angle.
- The other two sides that form the right angle are called legs (or catheti).
- Any right triangle can use the theorem, regardless of the letters used for sides.
Problem Solving Steps
- Begin by sketching the triangle and labeling all sides and corner points.
- Write out the Pythagorean theorem using side names or variables from the problem.
- To find a missing leg: (hypotenuse)^2 - (known leg)^2 = (missing leg)^2.
- Substitute the known values into the formula.
- Calculate the squares, subtract, and then take the square root to get the missing side length.
Example 1
- Triangle with sides: hypotenuse = 10 cm, one leg = 6 cm, find the other leg.
- Calculation: 10^2 - 6^2 = 100 - 36 = 64; sqrt(64) = 8 cm.
Example 2
- Triangle with sides: hypotenuse = 5 cm, one leg = 4 cm, find the other leg.
- Calculation: 5^2 - 4^2 = 25 - 16 = 9; sqrt(9) = 3 cm.
Key Terms & Definitions
- Right triangle — A triangle with one 90-degree angle.
- Hypotenuse — The side opposite the right angle; the longest side in a right triangle.
- Leg (Cathetus) — Either of the two sides that form the right angle in a right triangle.
- Square root — A value that, when multiplied by itself, gives the original number.
Action Items / Next Steps
- Practice sketching and labeling right triangles for different problems.
- Solve additional exercises calculating missing sides using the Pythagorean theorem.