Understanding the Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in Euclidean geometry that establishes a fixed mathematical relationship among the three sides of a right-angled triangle. Formulated by the ancient Greek mathematician Pythagoras, this theorem proves that the square of the longest side (the hypotenuse) is precisely equal to the sum of the squares of the other two shorter sides (the legs).
The Core Formula: \( a^2 + b^2 = c^2 \)
To utilize this theorem correctly, you must first assign the appropriate variables to the correct sides of your triangle:
- Variables \( a \) and \( b \): These represent the "legs" of the triangle. The legs are the two shorter sides that intersect at exactly 90 degrees to form the right angle.
- Variable \( c \): This always represents the "hypotenuse." The hypotenuse is mathematically guaranteed to be the longest side of a right triangle, and it is strictly positioned directly opposite the 90-degree right angle.
Solving for a Missing Hypotenuse (\( c \))
If you are given the lengths of both legs (\( a \) and \( b \)) and need to find the hypotenuse (\( c \)), the formula is algebraically rearranged to isolate \( c \). You square both \( a \) and \( b \), add them together, and then calculate the square root of that sum: \( c = \sqrt{a^2 + b^2} \).
Solving for a Missing Leg (\( a \) or \( b \))
If you are given the hypotenuse (\( c \)) and one leg (for example, \( b \)), you must manipulate the formula using subtraction. Square the hypotenuse, subtract the square of the known leg, and then take the square root of the result to find the missing leg: \( a = \sqrt{c^2 - b^2} \). It is critical to remember that the hypotenuse (\( c \)) must always be a larger number than either leg; otherwise, the triangle cannot physically exist.