Question

find the length of side x in simplest radical form with a rational denominator. answer attempt 1 out of 2 x = | submit answer

Explanation:

Step1: Identify the triangle type

The triangle is a right - isosceles triangle (two legs are equal, right - angled). Let the length of each leg be \(x\), and the hypotenuse \(c = 5\).
For a right - isosceles triangle, we use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), and since \(a = b=x\), the formula becomes \(x^{2}+x^{2}=c^{2}\), or \(2x^{2}=c^{2}\).

Step2: Substitute \(c = 5\) into the formula

Substitute \(c = 5\) into \(2x^{2}=c^{2}\), we get \(2x^{2}=5^{2}=25\).

Step3: Solve for \(x^{2}\)

Divide both sides of the equation \(2x^{2}=25\) by 2: \(x^{2}=\frac{25}{2}\).

Step4: Solve for \(x\)

Take the square root of both sides: \(x=\sqrt{\frac{25}{2}}\). To rationalize the denominator, we multiply the numerator and denominator inside the square root by 2: \(x = \sqrt{\frac{25\times2}{2\times2}}=\frac{\sqrt{50}}{2}=\frac{5\sqrt{2}}{2}\) (since \(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\)).

Answer:

\(\frac{5\sqrt{2}}{2}\)