Let's solve the problem step-by-step.
We are given that the diagonal of a TV is 26 inches long, and this forms a pair of 30-60-90 right triangles.
### Step-by-Step Solution:
1. Understanding the 30-60-90 Triangle Ratios:
- In a 30-60-90 right triangle, the ratios of the sides are fixed. Specifically:
- The side opposite the 30-degree angle (short leg) is [tex]\(x\)[/tex].
- The side opposite the 60-degree angle (long leg) is [tex]\(x\sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(2x\)[/tex].
2. Given Data:
- The diagonal of the TV, which is the hypotenuse of the 30-60-90 triangles formed, is 26 inches.
3. Finding the Short Leg (Opposite the 30-degree Angle):
- Since the hypotenuse is [tex]\(2x\)[/tex]:
- [tex]\(2x = 26\)[/tex]
- Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{26}{2} = 13
\][/tex]
4. Finding the Long Leg (Opposite the 60-degree Angle):
- The side opposite the 60-degree angle is [tex]\(x\sqrt{3}\)[/tex]:
- [tex]\(x = 13\)[/tex]
- So, the long leg is:
[tex]\[
13\sqrt{3}
\][/tex]
### Result of the Calculation:
- The length of the TV (opposite the 30-degree angle) is 13 inches.
- The width of the TV (opposite the 60-degree angle) is [tex]\(13\sqrt{3}\)[/tex] inches.
Given the options:
- A. [tex]\(52 \sqrt{2}\)[/tex] inches by [tex]\(52 \sqrt{2}\)[/tex] inches
- B. 52 inches by [tex]\(52 \sqrt{3}\)[/tex] inches
- C. [tex]\(13 \sqrt{2}\)[/tex] inches by [tex]\(13 \sqrt{2}\)[/tex] inches
- D. 13 inches by [tex]\(13 \sqrt{3}\)[/tex] inches
The correct answer is:
[tex]\[ \boxed{13 \text{ inches by } 13 \sqrt{3} \text{ inches}} \][/tex]
So the correct option is D.
