Answer :
To determine the length and width of the TV when given the diagonal of 28 inches, and knowing that it forms a pair of 30-60-90 right triangles, we need to use the properties of these triangles.
1. Understand the 30-60-90 Triangle Properties:
A 30-60-90 triangle has sides in a specific ratio:
- The side opposite the 30-degree angle is the shortest side and often called the short leg.
- The side opposite the 60-degree angle is longer and is related to the short leg.
- The hypotenuse is twice the length of the short leg.
The specific ratio of the sides for a 30-60-90 triangle is:
- Short leg (opposite 30°): [tex]\( x \)[/tex]
- Long leg (opposite 60°): [tex]\( x\sqrt{3} \)[/tex]
- Hypotenuse (opposite the right angle): [tex]\( 2x \)[/tex]
2. Identify the Given Information:
The diagonal of the TV, acting as the hypotenuse of the 30-60-90 triangle pair, is 28 inches.
3. Relate the Hypotenuse to the Other Sides:
From the properties of the triangle, the hypotenuse (diagonal) is twice the short leg (length):
[tex]\[ 2x = 28 \][/tex]
4. Solve for the Short Leg [tex]\( x \)[/tex]:
[tex]\[ x = \frac{28}{2} = 14 \][/tex]
Thus, the short leg (length) of the TV is 14 inches.
5. Determine the Long Leg (Width):
According to the triangle properties, the long leg (width) is:
[tex]\[ x\sqrt{3} = 14\sqrt{3} \][/tex]
6. Final Length and Width:
- Length: 14 inches
- Width: [tex]\( 14\sqrt{3} \)[/tex] inches
Hence, the exact length and width of the TV are:
[tex]\[ 14 \text{ inches by } 14\sqrt{3} \text{ inches} \][/tex]
Therefore, the correct answer is:
B. 14 inches by [tex]\( 14\sqrt{3} \)[/tex] inches
1. Understand the 30-60-90 Triangle Properties:
A 30-60-90 triangle has sides in a specific ratio:
- The side opposite the 30-degree angle is the shortest side and often called the short leg.
- The side opposite the 60-degree angle is longer and is related to the short leg.
- The hypotenuse is twice the length of the short leg.
The specific ratio of the sides for a 30-60-90 triangle is:
- Short leg (opposite 30°): [tex]\( x \)[/tex]
- Long leg (opposite 60°): [tex]\( x\sqrt{3} \)[/tex]
- Hypotenuse (opposite the right angle): [tex]\( 2x \)[/tex]
2. Identify the Given Information:
The diagonal of the TV, acting as the hypotenuse of the 30-60-90 triangle pair, is 28 inches.
3. Relate the Hypotenuse to the Other Sides:
From the properties of the triangle, the hypotenuse (diagonal) is twice the short leg (length):
[tex]\[ 2x = 28 \][/tex]
4. Solve for the Short Leg [tex]\( x \)[/tex]:
[tex]\[ x = \frac{28}{2} = 14 \][/tex]
Thus, the short leg (length) of the TV is 14 inches.
5. Determine the Long Leg (Width):
According to the triangle properties, the long leg (width) is:
[tex]\[ x\sqrt{3} = 14\sqrt{3} \][/tex]
6. Final Length and Width:
- Length: 14 inches
- Width: [tex]\( 14\sqrt{3} \)[/tex] inches
Hence, the exact length and width of the TV are:
[tex]\[ 14 \text{ inches by } 14\sqrt{3} \text{ inches} \][/tex]
Therefore, the correct answer is:
B. 14 inches by [tex]\( 14\sqrt{3} \)[/tex] inches
