Question

find a.
3
2√3
6
2√6

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the sides are in the ratio \(1:1:\sqrt{2}\). Let the legs be \(x\) (here \(b = 2\sqrt{3}\)) and the hypotenuse be \(x\sqrt{2}\).

Step2: Use Pythagorean theorem

In a right - triangle \(a^{2}=b^{2}+(2\sqrt{3})^{2}\), but since it's a 45 - 45 - 90 triangle \(a=\sqrt{2}\times b\). Substitute \(b = 2\sqrt{3}\) into the formula \(a=\sqrt{2}\times2\sqrt{3}\).
Using the property \(\sqrt{m}\times\sqrt{n}=\sqrt{mn}\), we have \(a = 2\sqrt{6}\).

Answer:

\(2\sqrt{6}\)