Question

given ( ep = fp ) and ( gq = fq ), what is the perimeter of ( \triangle efg )?

Explanation:

Step1: Use the mid - segment property

Since \(EP = FP\) and \(GQ = FQ\), \(PQ\) is the mid - segment of \(\triangle EFG\). By the mid - segment theorem, \(EG = 2PQ\). So \(4y + 2+4y + 4=2(x + 2y)\).
Also, since \(EP = FP\), \(2x=4y + 2\).
From \(2x=4y + 2\), we can express \(x = 2y+1\).
Substitute \(x = 2y + 1\) into \(4y + 2+4y + 4=2(x + 2y)\):

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This is wrong. Let's use the property of equal segments.
Since \(EP = FP\), \(2x=4y + 2\) (because \(EP = FP\)), so \(x = 2y + 1\).
Since \(GQ = FQ\), \(3x-1=4y + 4\).
Substitute \(x = 2y + 1\) into \(3x-1=4y + 4\):

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If \(y = 1\), then from \(x = 2y+1\), \(x=2\times1 + 1=3\).

Step2: Calculate the side lengths

\(EF=2x+4y + 2\), substitute \(x = 3\) and \(y = 1\): \(EF=2\times3+4\times1 + 2=6 + 4+2=12\).
\(FG=3x-1+4y + 4\), substitute \(x = 3\) and \(y = 1\): \(FG=3\times3-1+4\times1 + 4=9-1+4 + 4=16\).
\(EG=2(x + 2y)\), substitute \(x = 3\) and \(y = 1\): \(EG=2\times(3+2\times1)=10\).

Step3: Calculate the perimeter

The perimeter of \(\triangle EFG\) is \(P=EF + FG+EG\).
\(P=12 + 16+10=38\).

Answer:

\(38\)