The lengths of 3 sides of triangle ABC is a, b, c where a, b, c are integers and c is the base side.
We have \(c<a+b \implies 2c < a+b+c=30 \implies c<15\). Triangle ABC is built to maximize \(c\), so \(c=14\). The length of base side is 14.

For any triangle that has the same perimeter, equilateral triangle has maximum area. So triangle DEF has 3 sides \(d=e=f=10\). The length of base side is 10.

Hence, the ratio is 14 : 10 = 7 : 5

The answer is D.
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I use Heron's formula:

\(S_{ABC}=\sqrt{p(p-a)(p-b)(p-c)}\) where \(p=\frac{a+b+c}{2}=const\)

From inequation: \(xyz \leq ( \frac{x+y+z}{3})^3\) for \(x, y, z>0\). Sign \("="\) occurs \(\iff x=y=z\)

We have \(S_{ABC} \leq \sqrt{p \times \left ( \frac{p-a+p-b+p-c}{3} \right) ^3}= \sqrt{p \times (\frac{p}{3})^3}=\frac{p^2}{3\sqrt{3}}\)

\(maxS_{ABC}=\frac{p^2}{3\sqrt{3}} \iff a=b=c=\frac{2p}{3}\)
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Actual LSAT CR bank by Broall

How to solve quadratic equations - Factor quadratic equations
Factor table with sign: The useful tool to solve polynomial inequalities
Applying AM-GM inequality into finding extreme/absolute value

New Error Log with Timer

c < a + b => c + c < a + b+ c => 2c < a + b + c
_________________

Actual LSAT CR bank by Broall

How to solve quadratic equations - Factor quadratic equations
Factor table with sign: The useful tool to solve polynomial inequalities
Applying AM-GM inequality into finding extreme/absolute value

New Error Log with Timer