Let's break down the problem step-by-step:
1. Understanding the 30-60-90 Triangle:
In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the sides have a specific ratio: 1 (opposite the 30° angle) : [tex]\(\sqrt{3}\)[/tex] (opposite the 60° angle) : 2 (the hypotenuse).
2. Given:
We're given the length of the diagonal (which is the hypotenuse of the 30-60-90 triangle) as 26 inches.
3. Calculate the shorter side (opposite the 30° angle):
The hypotenuse is twice the length of the shorter side. Therefore, the shorter side is:
[tex]\[
\text{shorter side} = \frac{\text{hypotenuse}}{2} = \frac{26}{2} = 13 \text{ inches}
\][/tex]
4. Calculate the longer side (opposite the 60° angle):
The longer side in a 30-60-90 triangle is [tex]\(\sqrt{3}\)[/tex] times the shorter side. Therefore, the longer side is:
[tex]\[
\text{longer side} = 13 \times \sqrt{3}
\][/tex]
Thus, the exact length and width of the TV are 13 inches by [tex]\(13 \sqrt{3}\)[/tex] inches.
So, the correct answer is B. 13 inches by [tex]$13 \sqrt{3}$[/tex].
