Question

given the image, which proportion is true?

\\(\frac{ql}{pm} = \frac{sq}{jp}\\)

\\(\frac{kj}{jp} = \frac{ks}{sl}\\)

\\(\frac{jp}{kp} = \frac{lq}{kq}\\)

\\(\frac{kl}{lw} = \frac{rs}{mw}\\)

Explanation:

Identify parallel lines and transversals

The image shows three vertical parallel lines (indicated by the pink arrows): line \(IS\), line \(PQ\), and line \(ML\).
Two transversals intersect these parallel lines: transversal \(KM\) (with points \(J\), \(P\), \(M\)) and transversal \(KL\) (with points \(S\), \(Q\), \(L\)). Note that the point labeled \(I\) is on the first parallel line, but the transversal line passes through \(J\) (where \(J\) is the intersection of transversal \(KM\) and line \(IS\)). Thus, the points on transversal \(KM\) are \(K\), \(J\), \(P\), \(M\).
The points on transversal \(KL\) are \(K\), \(S\), \(Q\), \(L\).

Apply the Parallel Lines Proportionality Theorem

By the Parallel Lines Proportionality Theorem (or Three Parallel Lines Theorem), three parallel lines cut off proportional segments on any two transversals:

$$ \frac{JS}{PQ} = \frac{SQ}{QL} \quad \text{is not directly related, but along the transversals:} $$

$$ \frac{JP}{PM} = \frac{SQ}{QL} \implies \frac{QL}{PM} = \frac{SQ}{JP} $$

Let's check the first option:

$$ \frac{QL}{PM} = \frac{SQ}{JP} $$

This matches the proportional segments cut by the parallel lines \(IS\), \(PQ\), and \(ML\) on the two transversals:

• Segments on transversal \(KM\) between the parallel lines are \(JP\) and \(PM\).
• Corresponding segments on transversal \(KL\) between the parallel lines are \(SQ\) and \(QL\).

Therefore, the ratio of corresponding segments is equal:

$$ \frac{SQ}{JP} = \frac{QL}{PM} \implies \frac{QL}{PM} = \frac{SQ}{JP} $$

Verify other options

• Option 2: \(\frac{KJ}{JP} = \frac{KS}{SL}\). This is false because \(SL = SQ + QL\), whereas the corresponding segment to \(JP\) is \(SQ\).
• Option 3: \(\frac{JP}{KP} = \frac{LQ}{KQ}\). This is false because \(JP\) corresponds to \(SQ\), not \(LQ\).
• Option 4: \(\frac{KL}{LW} = \frac{RS}{MW}\). This involves segments on different lines that do not correspond proportionally.

Answer:

• (A) \(\frac{QL}{PM} = \frac{SQ}{JP}\) (Correct answer)
• (B) \(\frac{KJ}{JP} = \frac{KS}{SL}\)
• (C) \(\frac{JP}{KP} = \frac{LQ}{KQ}\)
• (D) \(\frac{KL}{LW} = \frac{RS}{MW}\)