Solving for the Lengths of Legs in an Isosceles Right-Angled Triangle
This image shows a right-angled triangle with one of the angles being \(45^\circ\), which makes it an isosceles right-angled triangle (since the other non-right angle must also be \(45^\circ\)). The hypotenuse of this triangle is given as 5 units.
In an isosceles right-angled triangle, the lengths of the two legs (sides opposite the \(45^\circ\) angles) are equal. If we let one of the legs be \(x\), we can use the Pythagorean theorem to solve for \(x\):
\[x^2 + x^2 = 5^2\]
\[2x^2 = 25\]
\[x^2 = \frac{25}{2}\]
\[x = \sqrt{\frac{25}{2}}\]
\[x = \frac{5}{\sqrt{2}}\]
Multiplying the numerator and denominator by \(\sqrt{2}\) to rationalize the denominator, we get:
\[x = \frac{5\sqrt{2}}{2}\]
So the lengths of the two legs are each \(\frac{5\sqrt{2}}{2}\) units.
