Overview
This lesson covers the basics of right-angle triangles, introduces Pythagoras' theorem, demonstrates how to use it to solve for unknown side lengths, and provides real-world examples.
Right Angle Triangles
- A right angle triangle has one angle of exactly 90 degrees.
- All triangles have three sides, but in right angle triangles, one side is special: the hypotenuse.
- The hypotenuse is always opposite the right angle and is the longest side in a right angle triangle.
Pythagoras’ Theorem
- Pythagoras’ theorem allows you to calculate the length of one side of a right angle triangle if the other two sides are known.
- The theorem is proven and works for all right angle triangles.
- The formula is: ( a^2 = b^2 + c^2 ), where ( a ) is the hypotenuse and ( b ), ( c ) are the other two sides.
- The hypotenuse must always be labeled ( a ) in the formula; ( b ) and ( c ) can be either of the other two sides.
Example Problems
- To find a missing side, identify the hypotenuse and substitute known values into ( a^2 = b^2 + c^2 ).
- If solving for a side other than the hypotenuse, rearrange: ( x^2 = a^2 - b^2 ).
- Use square roots to solve for the side length after substituting values.
Real World Applications
- Pythagoras’ theorem can determine unknown distances, such as the length of a cable attached to a building when given height and ground distance.
- Represent the scenario as a right angle triangle, identify the hypotenuse, and solve using the formula.
Key Terms & Definitions
- Right Angle Triangle — Triangle with one 90-degree angle.
- Hypotenuse — Side opposite the right angle; always the triangle’s longest side.
- Pythagoras’ Theorem — A mathematical formula: ( a^2 = b^2 + c^2 ) for right angle triangles.
Action Items / Next Steps
- Practice solving for unknown sides in right angle triangles using Pythagoras’ theorem.
- Use a calculator for square roots as needed.
- Review the properties of right angle triangles and the steps for identifying the hypotenuse.