Question

i 176b (4) x 45° 45° 2√6

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 legs are of equal length and the hypotenuse $c$ is $\sqrt{2}$ times the length of a leg $a$ (i.e., $c = a\sqrt{2}$).

Step2: Determine the length of the hypotenuse

The given leg length $a = 2\sqrt{6}$. Using the ratio for a 45 - 45-90 triangle, the hypotenuse $x=a\sqrt{2}$. Substitute $a = 2\sqrt{6}$ into the formula: $x=2\sqrt{6}\times\sqrt{2}$.

Step3: Simplify the expression

We know that $\sqrt{m}\times\sqrt{n}=\sqrt{mn}$. So, $2\sqrt{6}\times\sqrt{2}=2\sqrt{6\times2}=2\sqrt{12}$. Further simplify $\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}$. Then $2\sqrt{12}=2\times2\sqrt{3}=4\sqrt{3}$.

Answer:

$4\sqrt{3}$