Question

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

Explanation:

Step1: Identify Isosceles Trapezoid

Since \( EF \parallel GH \) (marked by arrows) and \( \angle E \cong \angle H \), \( EFGH \) is an isosceles trapezoid. Thus, \( EF = GH \) and \( EH \) is the base, so \( EF = GH \) implies \( 4n - 4 = 2n + 6 \).

Step2: Solve for \( n \)

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

Step3: Calculate \( GH \)

Substitute \( n = 5 \) into \( 3n + 3 \) (wait, no—wait, \( EF = 4n - 4 \), but \( GH \) is \( 3n + 3 \)? Wait, no, correction: Wait, in isosceles trapezoid, the non - parallel sides (legs) are equal? Wait, no, the diagram: \( EF \) and \( GH \) are the non - parallel? Wait, no, the arrows: \( EH \) is a base (parallel), and \( FG \) is parallel to \( EH \)? Wait, no, the arrows on \( FG \) and \( EH \) mean \( FG \parallel EH \), and the angles at \( E \) and \( H \) are equal, so \( EF = GH \) (legs of isosceles trapezoid). Wait, \( EF = 4n - 4 \), \( GH = 2n + 6 \)? No, the problem says \( GH \) is \( 3n + 3 \)? Wait, no, the diagram: \( FG \) is \( 3n + 3 \), \( EF \) is \( 4n - 4 \), \( GH \) is \( 2n + 6 \)? Wait, I made a mistake. Wait, the question is about \( GH \), which is \( 3n + 3 \)? Wait, no, let's re - examine. The line segment \( GH \) is labeled \( 3n + 3 \)? Wait, no, the diagram: \( F \) to \( G \) is \( 3n + 3 \), \( G \) to \( H \) is \( 2n + 6 \), \( E \) to \( F \) is \( 4n - 4 \), \( E \) to \( H \) is the base. Wait, no, the key is that since \( FG \parallel EH \) and \( \angle E=\angle H \), \( EFGH \) is an isosceles trapezoid, so \( EF = GH \). So \( 4n - 4=2n + 6 \), solved \( n = 5 \). Then \( GH \) is \( 3n + 3 \)? Wait, no, the problem's \( GH \) length: wait, the options are 7,5,16,24. Wait, if \( n = 5 \), then \( 3n+3=3(5)+3 = 18 \)? No, that's not an option. Wait, I messed up. Wait, maybe \( FG \) is parallel to \( EH \), and \( EF = GH \) (legs), so \( EF = 4n - 4 \), \( GH = 2n + 6 \), but also \( FG = 3n + 3 \). Wait, no, the question is "What is the length of line segment \( GH \)". Wait, maybe the legs are \( EF \) and \( GH \), so \( EF = GH \), so \( 4n - 4=2n + 6 \), \( n = 5 \). Then \( GH = 4n - 4=4(5)-4 = 16 \)? Wait, \( 4*5 - 4 = 16 \), which is one of the options (16 units). Ah, I see, I mislabeled the segments. So \( GH \) is equal to \( EF \), so \( EF = 4n - 4 \), and since \( EF = GH \), and we found \( n = 5 \), then \( GH = 4(5)-4=16 \).

Answer:

16 units