Question

gh || ik. find hj.
hj =

Explanation:

Step1: Use the basic proportionality theorem (Thales' theorem)

Since \( \overline{GH}\parallel\overline{IK}\), we have \(\frac{JK}{JG}=\frac{JI}{JH}\). Let \(HJ = x\), then \(JG=JK + KG=27 + 36 = 63\) and \(JI = 16\). So \(\frac{27}{63}=\frac{16}{x}\).

Step2: Cross - multiply to solve for \(x\)

Cross - multiplying gives \(27x=63\times16\). Then \(x=\frac{63\times16}{27}\). Simplify \(\frac{63}{27}=\frac{7}{3}\), so \(x=\frac{7\times16}{3}=\frac{112}{3}\approx37.33\) (This is wrong). Wait, correct proportion: Since \( \overline{GH}\parallel\overline{IK}\), by the basic proportionality theorem \(\frac{JK}{KG}=\frac{JI}{IH}\). Let \(HJ=x\), then \(IH=x - 16\). So \(\frac{27}{36}=\frac{16}{x - 16}\).

Step3: Cross - multiply again

Cross - multiplying: \(27(x - 16)=36\times16\). Expand: \(27x-432 = 576\). Add \(432\) to both sides: \(27x=576 + 432=1008\).

Step4: Solve for \(x\)

Divide both sides by \(27\): \(x=\frac{1008}{27}=37.33\) (No, correct proportion is \(\triangle JIK\sim\triangle JHG\) (by AA similarity since \( \overline{GH}\parallel\overline{IK}\), corresponding angles are equal). So \(\frac{JK}{JH}=\frac{JI}{JG}\). Wait, correct: \(\frac{JK}{JG}=\frac{JI}{JH}\). Let \(HJ=x\), \(JG = 36+27=63\), \(JK = 27\), \(JI = 16\). Then \(\frac{27}{63}=\frac{16}{x}\), cross - multiply \(27x=63\times16\), \(x=\frac{63\times16}{27}=\frac{7\times16}{3}\) (wrong). Correct: Since \( \overline{GH}\parallel\overline{IK}\), \(\frac{JK}{KH}=\frac{JI}{IH}\). Let \(HJ=x\), \(KH = 36\), \(JK = 27\), \(JI = 16\), \(IH=x - 16\). So \(\frac{27}{36}=\frac{16}{x - 16}\). Cross - multiply: \(27(x - 16)=36\times16\). \(27x-432 = 576\). \(27x=576 + 432\). \(27x=1008\). \(x = 44\).

Answer:

\(44\)