To find the exact length and width of the TV with a diagonal of 28 inches that forms a pair of 30-60-90 right triangles, we need to use the properties of 30-60-90 triangles. In this special right triangle, the sides are in the ratio 1 : √3 : 2.
Let's denote:
- The hypotenuse (diagonal) as [tex]\( d = 28 \)[/tex] inches,
- The shortest side (half the length of the TV) as [tex]\( a \)[/tex],
- The longer side (half the height of the TV) as [tex]\( b \)[/tex].
Step-by-Step Solution:
1. Identify the ratios in a 30-60-90 triangle:
The sides are in the ratio:
- Hypotenuse = 2,
- Short side = 1,
- Long side = √3.
2. Determine the short side [tex]\( a \)[/tex]:
Since the hypotenuse of the triangle (which is the diagonal of the TV) is 28 inches, and the hypotenuse in the ratio is 2, we can write:
[tex]\[
\frac{a}{1} = \frac{d}{2}
\][/tex]
Therefore:
[tex]\[
a = \frac{d}{2} = \frac{28}{2} = 14 \text{ inches}
\][/tex]
3. Determine the long side [tex]\( b \)[/tex]:
Using the same ratio logic:
[tex]\[
\frac{b}{\sqrt{3}} = \frac{d}{2}
\][/tex]
Therefore:
[tex]\[
b = \frac{d \times \sqrt{3}}{2} = \frac{28 \times \sqrt{3}}{2} = 14 \sqrt{3} \text{ inches}
\][/tex]
4. Calculate the length and width of the TV:
Since the TV diagonal forms two such triangles, the full length and width of the TV will be:
- Length (2 times the short side):
[tex]\[
\text{Length} = 2a = 2 \times 14 = 28 \text{ inches}
\][/tex]
- Width (2 times the long side):
[tex]\[
\text{Width} = 2b = 2 \times 14\sqrt{3} = 28\sqrt{3} \text{ inches}
\][/tex]
Now, we compare these dimensions with the options given:
A. [tex]\( 14 \sqrt{2} \)[/tex] inches by [tex]\( 14 \sqrt{2} \)[/tex] inches.
B. 56 inches by [tex]\( 56 \sqrt{3} \)[/tex] inches.
C. 14 inches by [tex]\( 14 \sqrt{3} \)[/tex] inches.
D. [tex]\( 56 \sqrt{2} \)[/tex] inches by [tex]\( 56 \sqrt{2} \)[/tex] inches.
Considering the actual length and width we calculated:
- Length = 14 inches
- Width = 14√3 inches.
Therefore, the correct answer is: C. 14 inches by 14√3 inches.
