Two transversals S and T cut a set of distinct parallel lines. S cuts the parallel lines in points A, B, C, D and T cuts the parallel lines in points E,F, G and H, respectivley. If AB = 4, CD = 3 and EF = 12, then what is the length of GH ?

Options

A. 4
B. 6
C. 8
D. 9

Answer

To solve the problem, we will use the concept of proportionality of segments created by transversals cutting through parallel lines.

### Step-by-Step Solution:

1. **Identify the segments**: We have two transversals S and T cutting through parallel lines. The segments formed by transversal S are AB and CD, and by transversal T are EF and GH. We know:
- \( AB = 4 \)
- \( CD = 3 \)
- \( EF = 12 \)
- \( GH = x \) (unknown)

2. **Set up the proportion**: According to the property of transversals cutting parallel lines, the ratios of the lengths of segments on one transversal are equal to the ratios of the lengths of segments on the other transversal. Thus, we can set up the following proportion:
\[ \frac{AB}{CD} = \frac{EF}{GH} \]

3. **Substitute the known values**: Plugging in the values we have:
\[ \frac{4}{3} = \frac{12}{x} \]

4. **Cross-multiply to solve for x**: Cross-multiplying gives us:
\[ 4x = 3 \times 12 \]
\[ 4x = 36 \]

5. **Divide by 4**: To find \( x \), divide both sides by 4:
\[ x = \frac{36}{4} = 9 \]

6. **Conclusion**: Therefore, the length of segment GH is:
\[ GH = 9 \]

### Final Answer:
The length of GH is 9.

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