Question

solve for x. the triangles in each pair are similar.
\\(\triangle qrs \sim \triangle jkl\\)
triangle 1: vertices \\(q, r, s\\). side \\(qr = 5x - 5\\), side \\(qs = 20\\).
triangle 2: vertices \\(j, k, l\\). side \\(jk = 15\\), side \\(jl = 24\\).
options:
6
13
3
11

Explanation:

Identify corresponding sides of similar triangles

Given \(\triangle QRS \sim \triangle JKL\), the corresponding sides are:

$$ \frac{QR}{JK} = \frac{RS}{KL} = \frac{QS}{JL} $$

From the image:

• \(QR = 5x - 5\)
• \(QS = 24\)
• \(JK = 15\)
• \(JL = 24\)

Set up the proportion

$$ \frac{QR}{JK} = \frac{QS}{JL} \implies \frac{5x - 5}{15} = \frac{24}{24} $$

Solve for x

$$
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$$

Wait, let's re-examine the labels in the image carefully.
The labels are:

• Triangle 1: \(\triangle QRS\). Side \(QR = 5x - 5\), side \(QS = 20\) (or is it 20? Let's look closer. It says "20" at the bottom of the first triangle, so \(QS = 20\)).
• Triangle 2: \(\triangle JKL\). Side \(JK = 15\), side \(JL = 12\) (or is it 12? Let's look closer. It says "12" at the bottom left, and "15" on the left side. Let's re-verify the similarity statement: \(\triangle QRS \sim \triangle JKL\)).

If \(\triangle QRS \sim \triangle JKL\):

• \(QR\) corresponds to \(JK\).
• \(QS\) corresponds to \(JK\)? No, \(QS\) corresponds to \(JL\).

Let's look at the numbers:
First triangle: vertical-ish side \(QR = 5x - 5\), bottom side \(QS = 20\).
Second triangle: left side \(JK = 15\), bottom-left side \(JL = 12\) (or is it \(JK = 15\), \(JL = 12\)? Yes, the label on the bottom is 12, and the label on the left is 15).
Let's set up the ratio:

$$ \frac{QR}{JK} = \frac{QS}{JL} \implies \frac{5x - 5}{15} = \frac{20}{12} $$

Simplify \(\frac{20}{12}\):

$$ \frac{20}{12} = \frac{5}{3} $$

Now solve:

$$
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This matches the option "6" (the yellow button).

Answer:

• (A) 6 (Correct answer)
• (B) 13
• (C) 3
• (D) 11