To find the exact length and width of the TV with a given diagonal length of 26 inches, where the diagonal forms a pair of 30-60-90 right triangles, follow these steps:
1. Understanding the 30-60-90 triangle properties: In a 30-60-90 triangle, the sides follow a specific ratio. The ratio of the lengths of the sides opposite the 30°, 60°, and 90° angles is [tex]\(1 : \sqrt{3} : 2\)[/tex].
2. Diagonal as Hypotenuse: The diagonal of the TV is the hypotenuse of the overall right triangle, which can be considered as the hypotenuse of two 30-60-90 triangles put together. Therefore, it is twice the side opposite the 30° angle.
3. Calculating the shortest side (length):
- The length (shortest side) is opposite the 30° angle in a 30-60-90 triangle.
- Since the hypotenuse (diagonal) is 26 inches, the length opposite the 30° angle (half the hypotenuse of the whole triangle) is:
[tex]\[
\text{Length} = \frac{\text{Hypotenuse}}{2} = \frac{26}{2} = 13 \text{ inches}
\][/tex]
4. Calculating the longer side (width):
- The width (longer side) is opposite the 60° angle in a 30-60-90 triangle.
- Given the side opposite the 30° angle (length obtained above) is 13 inches, this can be used to find the width using the ratio [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, the width is:
[tex]\[
\text{Width} = 13 \times \sqrt{3} \text{ inches}
\][/tex]
Therefore, the exact length and width of the TV are 13 inches by [tex]\(13 \sqrt{3}\)[/tex] inches.
The correct answer is:
A. 13 inches by [tex]\(13 \sqrt{3}\)[/tex] inches
