Question

find the length of side x in simplest radical form with a rational denominator.
(there is a right triangle with a 45° angle, hypotenuse length 1, and side x opposite the other 45° angle (since its a right triangle with one 45° angle, its an isosceles right triangle).)
answer attempt 1 out of 2
x - input box
√

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the ratio of the sides is \(1:1:\sqrt{2}\), where the hypotenuse \(c\) and the legs \(a\) and \(b\) satisfy \(c = a\sqrt{2}=b\sqrt{2}\). Here, the hypotenuse \(c = 1\) and the leg \(x\) (one of the non - hypotenuse sides).

Step2: Use the side - ratio formula

From \(c=a\sqrt{2}\), we can solve for \(a\) (where \(a=x\)). Rearranging the formula gives \(x=\frac{c}{\sqrt{2}}\).

Step3: Rationalize the denominator

Multiply the numerator and denominator by \(\sqrt{2}\): \(x=\frac{1\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{2}}{2}\).

Answer:

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