Question

The triangle is an equilateral find the length of side x in simplest radical form with a rational denominator. The triangle is split in half making two right triangles and the hypotenuse of onside is the square root of 10. find x which is adjacent too the 90 degree angle.

Answer 1

To find the length of side \( x \) in an equilateral triangle that has been split in half, we start with the following information:

1. The equilateral triangle has a hypotenuse of \( \sqrt{10} \) when split in half.
2. The half of an equilateral triangle creates two 30-60-90 right triangles.

In a 30-60-90 triangle, the ratios of the sides are as follows:
- The side opposite the 30-degree angle (short leg) is \( x \).
- The side opposite the 60-degree angle (long leg) is \( x\sqrt{3} \).
- The hypotenuse is \( 2x \).

Given that the hypotenuse is \( \sqrt{10} \), we can set up the equation:

\[
2x = \sqrt{10}
\]

To solve for \( x \), we divide both sides by 2:

\[
x = \frac{\sqrt{10}}{2}
\]

Now, to write this in simplest radical form with a rational denominator, we have:

\[
x = \frac{\sqrt{10}}{2}
\]

This expression is already in the required form, and the denominator is rational.

Thus, the length of side \( x \) is

\[
\boxed{\frac{\sqrt{10}}{2}}
\]