Question

fill in the blank 1 point
find gh.

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment of a triangle (a segment connecting the mid - points of two sides of a triangle) is half the length of the third side. Here, \(GH\) is the mid - segment of \(\triangle DEF\), so \(GH=\frac{1}{2}DF\). Also, \(EG = GD\) (given by the marks on the sides), and \(EH=HF\) (given by the marks on the sides). So, \(3x - 4=\frac{1}{2}(9x - 59)\).

Step2: Solve the equation for \(x\)

Multiply both sides of the equation \(3x - 4=\frac{1}{2}(9x - 59)\) by \(2\) to get rid of the fraction:

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Subtract \(6x\) from both sides: \(-8 = 3x-59\).
Add \(59\) to both sides: \(3x=59 - 8=51\).
Divide both sides by \(3\): \(x = 17\).

Step3: Calculate the length of \(GH\)

Substitute \(x = 17\) into the expression for \(GH\). Since \(GH=3x - 4\), then \(GH=3\times17-4\).

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Answer:

\(GH = 47\)