Question

find q.
write your answer as a whole number or a decimal. do not round.
q = \boxed{} yards

Explanation:

Step1: Identify similar triangles

Since \(\angle I\cong\angle L\), \(\angle KIJ\cong\angle MLJ\) (vertical angles), by the AA (Angle - Angle) similarity criterion, \(\triangle IKJ\sim\triangle LMJ\).

Step2: Set up proportion

For similar triangles \(\triangle IKJ\) and \(\triangle LMJ\), the ratios of corresponding sides are equal. So, \(\frac{IK}{ML}=\frac{KJ}{MJ}\).
Substitute \(IK = 72\), \(ML=96\), \(KJ = 66\), and \(MJ=q\) into the proportion: \(\frac{72}{96}=\frac{66}{q}\).

Step3: Cross - multiply

Cross - multiplying gives \(72q=96\times66\).
Calculate \(96\times66 = 6336\), so \(72q = 6336\).

Step4: Solve for \(q\)

Divide both sides by 72: \(q=\frac{6336}{72}\).
\(q = 88\).

Answer:

\(88\)