Answer :
To determine the length and width of the TV based on the given diagonal, we can use properties of 30-60-90 right triangles. Here's the step-by-step solution:
1. Understand the properties of a 30-60-90 triangle:
- In a 30-60-90 triangle, the sides opposite the 30°, 60°, and 90° angles have a specific ratio of lengths: 1 : [tex]\(\sqrt{3}\)[/tex] : 2.
- The side opposite the 30° angle is the shortest side.
- The side opposite the 60° angle is [tex]\(\sqrt{3}\)[/tex] times the shortest side.
- The side opposite the 90° angle (the hypotenuse) is twice the shortest side.
2. Given information:
- The diagonal (which is the hypotenuse in our 30-60-90 triangle) is 30 inches long.
3. Solve for the shortest side:
- Since the hypotenuse is twice the shortest side, we set up the equation:
[tex]\[ \text{Hypotenuse} = 2 \times \text{Shortest side} \][/tex]
Substituting the given hypotenuse:
[tex]\[ 30 = 2 \times \text{Shortest side} \][/tex]
Solving for the shortest side:
[tex]\[ \text{Shortest side} = \frac{30}{2} = 15 \text{ inches} \][/tex]
4. Solve for the longer side:
- The longer side is [tex]\(\sqrt{3}\)[/tex] times the shortest side:
[tex]\[ \text{Longer side} = 15 \times \sqrt{3} \][/tex]
From this analysis we determine that:
- The length (shorter side) is 15 inches.
- The width (longer side) is [tex]\(15 \sqrt{3}\)[/tex] inches.
Thus, the correct answer is:
[tex]\[ \boxed{\text{B. } 15 \text{ inches by } 15 \sqrt{3} \text{ inches}} \][/tex]
1. Understand the properties of a 30-60-90 triangle:
- In a 30-60-90 triangle, the sides opposite the 30°, 60°, and 90° angles have a specific ratio of lengths: 1 : [tex]\(\sqrt{3}\)[/tex] : 2.
- The side opposite the 30° angle is the shortest side.
- The side opposite the 60° angle is [tex]\(\sqrt{3}\)[/tex] times the shortest side.
- The side opposite the 90° angle (the hypotenuse) is twice the shortest side.
2. Given information:
- The diagonal (which is the hypotenuse in our 30-60-90 triangle) is 30 inches long.
3. Solve for the shortest side:
- Since the hypotenuse is twice the shortest side, we set up the equation:
[tex]\[ \text{Hypotenuse} = 2 \times \text{Shortest side} \][/tex]
Substituting the given hypotenuse:
[tex]\[ 30 = 2 \times \text{Shortest side} \][/tex]
Solving for the shortest side:
[tex]\[ \text{Shortest side} = \frac{30}{2} = 15 \text{ inches} \][/tex]
4. Solve for the longer side:
- The longer side is [tex]\(\sqrt{3}\)[/tex] times the shortest side:
[tex]\[ \text{Longer side} = 15 \times \sqrt{3} \][/tex]
From this analysis we determine that:
- The length (shorter side) is 15 inches.
- The width (longer side) is [tex]\(15 \sqrt{3}\)[/tex] inches.
Thus, the correct answer is:
[tex]\[ \boxed{\text{B. } 15 \text{ inches by } 15 \sqrt{3} \text{ inches}} \][/tex]
