Solving for the Length of a Right Triangle Side
The image displays a right-angled triangle with one leg labeled as 12, the hypotenuse labeled as 26, and the other leg labeled as \(x\). To solve for \(x\), we can use the Pythagorean theorem, which states for a right-angled triangle that \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse.
Given:
\(a = x\) (the unknown side we are trying to find)
\(b = 12\) (one of the legs)
\(c = 26\) (the hypotenuse)
The equation becomes:
\(x^2 + 12^2 = 26^2\)
Now we can solve for \(x\):
\(x^2 + 144 = 676\)
Subtract 144 from both sides to isolate \(x^2\):
\(x^2 = 676 - 144\)
\(x^2 = 532\)
Now take the square root of both sides to solve for \(x\):
\(x = \sqrt{532}\)
\(x\) is the square root of 532, which can be simplified:
\(x = \sqrt{4 \cdot 133}\)
\(x = \sqrt{4} \cdot \sqrt{133}\)
\(x = 2 \cdot \sqrt{133}\)
So the length of the unknown side \(x\) is \(2\sqrt{133}\), which is an exact value. If a decimal value is required, you would need to use a calculator to approximate the square root of 133.
