Question

the triangle below is isosceles. find the length of side x in simplest radical form with a rational denominator. \\(\sqrt{10}\\) (triangle with right angle and side x)

Explanation:

Step1: Identify equal sides

This is a right isosceles triangle, so the two legs are equal. Each leg has length $\sqrt{10}$.

Step2: Apply Pythagorean theorem

Use $a^2 + b^2 = c^2$, where $a=b=\sqrt{10}$, $c=x$.
$$x^2 = (\sqrt{10})^2 + (\sqrt{10})^2$$

Step3: Calculate squared terms

Compute each squared radical: $(\sqrt{10})^2=10$.
$$x^2 = 10 + 10 = 20$$

Step4: Solve for x and rationalize

Take square root, simplify $\sqrt{20}$:
$$x = \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$

Answer:

$2\sqrt{5}$