What is the Pythagoras Theorem?
The Pythagoras Theorem is a fundamental geometric rule that describes the unique relationship between the three sides of any right-angled triangle. A right-angled triangle is a triangle in which exactly one angle measures 90 degrees. The theorem states that the square of the length of the longest side is exactly equal to the sum of the squares of the lengths of the other two sides.
The sides of a right-angled triangle have specific names:
- Hypotenuse: The longest side of the triangle, which always sits directly opposite the 90-degree right angle.
- Base: The horizontal side resting at the bottom of the triangle.
- Perpendicular: The straight upright side that forms a right angle with the base.
Imagine you are sliding down a straight playground slide. The ladder you climbed up is the perpendicular, the flat ground underneath between the ladder and the end of the slide is the base, and the long sloped ramp you slide down is the hypotenuse.
If we represent the lengths of the perpendicular and base as a and b, and the hypotenuse length as c, the formula is written as:
To prove this theorem, we use geometric similarity. By drawing a line from the right-angled corner straight down perpendicular to the hypotenuse, we split the large triangle into two smaller triangles. Both of these smaller triangles share the exact same structural proportions as the original large triangle, allowing us to use cross-multiplication ratios to verify that the squared areas must perfectly match.
| Side Name | Diagram Variable Label | Algebraic Formula Element |
|---|---|---|
| Base | b | Second squared leg (b²) |
| Hypotenuse | c | Longest isolated squared side (c²) |