Question

gh || fi. find hj.
hj =

Explanation:

Step1: Apply the Basic Proportionality Theorem (Thales' Theorem)

Since \( \overline{GH}\parallel\overline{FI}\), by the Basic Proportionality Theorem, we have the proportion \(\frac{JF}{FG}=\frac{JI}{IH}\). Let \(HJ = x\), then \(IH=x - 15\) and \(FG = 16\), \(JF=46 - 16=30\), \(JI = 15\).

Step2: Set up the proportion equation

\(\frac{30}{16}=\frac{15}{x - 15}\). Cross - multiply: \(30(x - 15)=16\times15\).

Step3: Expand and solve for \(x\)

Expand \(30(x - 15)\): \(30x-450 = 240\). Add \(450\) to both sides: \(30x=240 + 450=690\). Divide both sides by \(30\): \(x=\frac{690}{30}=23\).

Answer:

\(23\)