Pythagorean Theorem Formula
The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. According to this theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship is useful for calculating the unknown length of a side when the lengths of the other two sides are known.
Formula for the Pythagorean Theorem
If \( a \) and \( b \) are the lengths of the legs of a right triangle, and \( c \) is the length of the hypotenuse, then the Pythagorean Theorem is given by the formula:
\( c^2 = a^2 + b^2 \)
In this formula:
- \( c \) is the hypotenuse, the side opposite the right angle
- \( a \) and \( b \) are the legs of the right triangle, adjacent to the right angle
Detailed Explanation of the Formula
The Pythagorean Theorem formula \( c^2 = a^2 + b^2 \) is used to find the length of one side of a right triangle when the lengths of the other two sides are known. By rearranging the formula, we can solve for any unknown side:
- To find the hypotenuse: \( c = \sqrt{a^2 + b^2} \)
- To find a leg when \( c \) and \( a \) are known: \( b = \sqrt{c^2 – a^2} \)
- To find a leg when \( c \) and \( b \) are known: \( a = \sqrt{c^2 – b^2} \)
Example 1: Calculating the Hypotenuse
Problem: A right triangle has legs of lengths \( a = 3 \, \text{cm} \) and \( b = 4 \, \text{cm} \). Find the length of the hypotenuse \( c \).
Solution:
- Write down the formula: \( c = \sqrt{a^2 + b^2} \).
- Substitute \( a = 3 \) and \( b = 4 \): \( c = \sqrt{3^2 + 4^2} \).
- Calculate \( a^2 \) and \( b^2 \): \( c = \sqrt{9 + 16} \).
- Add the squares: \( c = \sqrt{25} \).
- Take the square root: \( c = 5 \, \text{cm} \).
The length of the hypotenuse is \( 5 \, \text{cm} \).
Example 2: Calculating a Leg
Problem: A right triangle has a hypotenuse of length \( c = 13 \, \text{cm} \) and one leg of length \( a = 5 \, \text{cm} \). Find the length of the other leg \( b \).
Solution:
- Write down the formula: \( b = \sqrt{c^2 – a^2} \).
- Substitute \( c = 13 \) and \( a = 5 \): \( b = \sqrt{13^2 – 5^2} \).
- Calculate \( c^2 \) and \( a^2 \): \( b = \sqrt{169 – 25} \).
- Subtract: \( b = \sqrt{144} \).
- Take the square root: \( b = 12 \, \text{cm} \).
The length of the other leg is \( 12 \, \text{cm} \).
The Pythagorean Theorem is widely used in geometry, trigonometry, and various applications in physics and engineering to calculate distances and determine unknown side lengths in right triangles.