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Which set of values could be the side lengths of a 30-60-90 triangle?

A. [tex]$\{6, 12, 12 \sqrt{3}\}$[/tex]
B. [tex]$\{6, 6 \sqrt{3}, 12\}$[/tex]
C. [tex]$\{6, 6 \sqrt{2}, 12\}$[/tex]
D. [tex]$\{6, 12, 12 \sqrt{2}\}$[/tex]

Sagot :

GinnyAnswer
To determine which set of values could be the side lengths of a 30-60-90 triangle, we need to understand the properties of a 30-60-90 triangle. In such a triangle, the sides are in a specific ratio:

- The side opposite the 30-degree angle is the shortest and is denoted as \( x \).
- The side opposite the 60-degree angle is \( x \sqrt{3} \).
- The hypotenuse (opposite the 90-degree angle) is \( 2x \).

Given these properties, we will now check each set of values to see which one fits the \( 1 : \sqrt{3} : 2 \) ratio.

### Option A: \(\{6, 12, 12\sqrt{3}\}\)

1. Let's denote the shortest side as \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \). Here it is given as \( 12 \), which does not fit.
3. The hypotenuse should be \( 12 \), and here it is \( 12\sqrt{3} \), which does not fit either.

So, Option A is not correct.

### Option B: \(\{6, 6 \sqrt{3}, 12\}\)

1. Let the shortest side be \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \), which matches the given value \( 6 \sqrt{3} \).
3. The hypotenuse should be \( 12 \), and it is given as \( 12 \), which also matches.

So, Option B fits the required side lengths of a 30-60-90 triangle.

### Option C: \(\{6, 6 \sqrt{2}, 12\}\)

1. Let the shortest side be \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \), but here it is given as \( 6 \sqrt{2} \), which does not fit.
3. The hypotenuse should be \( 12 \), and it matches, but the middle side is incorrect.

So, Option C is not correct.

### Option D: \(\{6, 12, 12 \sqrt{2}\}\)

1. Let the shortest side be \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \), but here it is given as \( 12 \), which does not fit.
3. The hypotenuse should be \( 12 \), and it is \( 12 \sqrt{2} \), which also does not fit.

So, Option D is not correct.

### Conclusion

The only set of values that matches the side lengths of a 30-60-90 triangle is Option B: \(\{6, 6 \sqrt{3}, 12\}\).

So, the correct answer is:

B. [tex]\( \{6, 6 \sqrt{3}, 12\} \)[/tex]
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