Question

the sides of a triangle measure $3\sqrt{6}$, $2\sqrt{24}$ and $7\sqrt{54}$, what is the perimeter of the triangle?\
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a. $31\sqrt{3}$\
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b. $19\sqrt{6}$\
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c. $24\sqrt{81}$\
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d. $28\sqrt{6}$\
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e. $12\sqrt{84}$

Explanation:

Step1: Simplify each square root term

First, simplify \(2\sqrt{24}\) and \(7\sqrt{54}\).

For \(2\sqrt{24}\):
We know that \(24 = 4\times6\), so \(\sqrt{24}=\sqrt{4\times6} = \sqrt{4}\times\sqrt{6}=2\sqrt{6}\). Then \(2\sqrt{24}=2\times2\sqrt{6} = 4\sqrt{6}\).

For \(7\sqrt{54}\):
We know that \(54 = 9\times6\), so \(\sqrt{54}=\sqrt{9\times6}=\sqrt{9}\times\sqrt{6} = 3\sqrt{6}\). Then \(7\sqrt{54}=7\times3\sqrt{6}=21\sqrt{6}\).

The first term is \(3\sqrt{6}\).

Step2: Find the perimeter

The perimeter \(P\) of a triangle is the sum of its three sides. So we add the three simplified terms:

\(P=3\sqrt{6}+4\sqrt{6}+21\sqrt{6}\)

Since the terms have the same radical part (\(\sqrt{6}\)), we can combine the coefficients:

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

Answer:

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