Question

if $m\angle efh = m\angle gfh = 54^\circ$, $eh = 3v$, and $gh = v + 48$, what is $gh$?
$gh = \square$

Explanation:

Step1: Identify congruent triangles

Since \( \angle EFH=\angle GFH = 54^\circ \), \( \angle HEF=\angle HGF = 90^\circ \), and \( FH \) is common, triangles \( \triangle EFH \) and \( \triangle GFH \) are congruent (AAS congruence). Thus, \( EH = GH \).

Step2: Set up equation

Given \( EH = 3v \) and \( GH=v + 48 \), set \( 3v=v + 48 \).

Step3: Solve for \( v \)

Subtract \( v \) from both sides: \( 3v - v=v + 48 - v \), so \( 2v = 48 \). Divide by 2: \( v=\frac{48}{2}=24 \).

Step4: Find \( GH \)

Substitute \( v = 24 \) into \( GH=v + 48 \): \( GH=24 + 48 = 72 \).

Answer:

72