Question

consider quadrilateral efgh. what is the length of line segment gh? 5 units 7 units 16 units 24 units

Explanation:

Step1: Identify the type of quadrilateral

Since \(EF\parallel GH\) and \(\angle E=\angle H\), quadrilateral \(EFGH\) is an isosceles trapezoid. In an isosceles trapezoid, the non - parallel sides (legs) are equal. So, \(EF = GH\), and also \(4n-4=2n + 6\).

Step2: Solve the equation for \(n\)

Subtract \(2n\) from both sides of the equation \(4n-4=2n + 6\):
\(4n-2n-4=2n-2n + 6\)
\(2n-4=6\)
Add \(4\) to both sides:
\(2n-4 + 4=6+4\)
\(2n=10\)
Divide both sides by \(2\): \(n = 5\).

Step3: Find the length of \(GH\)

Substitute \(n = 5\) into the expression for \(GH\). Given \(GH=3n + 3\), then \(GH=3\times5+3\).
First, calculate \(3\times5=15\), then \(15 + 3=18\) (This is wrong. Wait, no, we made a mistake above. Wait, in an isosceles trapezoid, the non - parallel sides (the legs \(EF\) and \(GH\) are not. Wait, no, in an isosceles trapezoid, the non - parallel sides (the non - base sides) are equal. The bases are \(FG\) and \(EH\). Wait, no, looking at the figure (assuming standard notation for trapezoid with \(FG\parallel EH\)), the non - parallel sides are \(EF\) and \(GH\). Wait, no, no! Wait, the problem is that in an isosceles trapezoid, the non - parallel sides (the legs) are equal. So \(EF = GH\) is wrong. Wait, no, no! Wait, the correct property: In an isosceles trapezoid, the non - parallel sides (the legs) are equal. So \(EF=GH\) is wrong. Wait, no, looking at the problem again. Wait, the problem is that \(EF\) and \(GH\) are the non - parallel sides. Wait, no, no! Wait, the formula for the sides: If \(EF\) and \(GH\) are the non - parallel sides (legs) of the trapezoid. Wait, no, the problem is that we confused the sides. Wait, actually, in an isosceles trapezoid \(EFGH\) with \(FG\parallel EH\), the legs are \(EF\) and \(GH\). So \(EF = GH\). But \(EF=4n - 4\) and \(GH = 2n+6\). So \(4n-4=2n + 6\). Solving \(4n-2n=6 + 4\), \(2n=10\), \(n = 5\). Then \(FG=3n+3\). Substitute \(n = 5\) into \(FG\): \(FG=3\times5+3=15 + 3=18\) (wrong). Wait, no! Wait, the problem asks for \(GH\). Wait, no, we made a mistake in variable assignment. Wait, no, if \(EF\) and \(GH\) are the legs (non - parallel sides) of the isosceles trapezoid, then \(EF = GH\). But \(EF = 4n-4\) and \(GH=2n + 6\). Solving \(4n-4=2n+6\) gives \(n = 5\). Then \(GH=2n+6=2\times5+6=10 + 6=16\).

Answer:

16 units