Question

The sides of a triangle measure 3 StartRoot 6 EndRoot, 2 StartRoot 24 EndRoot and 7 StartRoot 54 EndRoot. What is the perimeter of the triangle?

Answer options with 5 options
A.
31 StartRoot 2 EndRoot

B.
17 StartRoot 6 EndRoot

C.
24 StartRoot 21 EndRoot

D.
28 StartRoot 6 EndRoot

E.
12 StartRoot 84 EndRoot

Answer 1

To find the perimeter of the triangle, we need to first simplify each side:

1. **First side:**
\[
3\sqrt{6}
\]

2. **Second side:**
\[
2\sqrt{24} = 2\sqrt{4 \times 6} = 2 \cdot 2\sqrt{6} = 4\sqrt{6}
\]

3. **Third side:**
\[
7\sqrt{54} = 7\sqrt{9 \times 6} = 7 \cdot 3\sqrt{6} = 21\sqrt{6}
\]

Now, let's add all the sides together to find the perimeter:

\[
3\sqrt{6} + 4\sqrt{6} + 21\sqrt{6}
\]

Combining the terms:

\[
(3 + 4 + 21)\sqrt{6} = 28\sqrt{6}
\]

Thus, the perimeter of the triangle is:

\[
\boxed{28\sqrt{6}}
\]

Looking at the answer options, the correct choice is:

D. \(28 \sqrt{6}\)