Question

if $\angle egf \cong \angle egh$, $ef = t$, and $eh = 3t - 42$, what is $eh$?
image of a geometric figure with right angles at f and h, vertices g, e, f, h
$eh = \square$
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Explanation:

Step1: Identify Congruent Segments

Since \(\angle EGF \cong \angle EGH\), and \(EF \perp GF\), \(EH \perp GH\) (right angles), by the Angle - Bisector Theorem (or the property of angle - bisector and perpendiculars), \(EF = EH\).

Step2: Set Up Equation

We know \(EF=t\) and \(EH = 3t - 42\). Since \(EF = EH\), we set up the equation \(t=3t - 42\).

Step3: Solve for \(t\)

Subtract \(t\) from both sides: \(0 = 3t-t - 42\), which simplifies to \(0 = 2t-42\). Then add 42 to both sides: \(2t=42\). Divide both sides by 2: \(t = 21\).

Step4: Find \(EH\)

Substitute \(t = 21\) into the expression for \(EH\): \(EH=3t - 42=3\times21-42=63 - 42 = 21\).

Answer:

\(21\)