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In this problem, we are given a triangle with lines \(PQ \parallel RS\), which means the triangles \( \triangle PQR \) and \( \triangle PQT \) are similar due to the basic property of parallel lines in triangles (corresponding angles are equal).

We know the following lengths:

- \( RP = 7 \, \text{cm} \)
- \( PT = 35 \, \text{cm} \)
- \( SQ = 13 \, \text{cm} \)

The question asks us to find the length of \( QT \).

### Step-by-step solution:

Since \(PQ \parallel RS\), the triangles \( \triangle PQT \) and \( \triangle PRS \) are similar by the Basic Proportionality Theorem (or Thales' theorem). This gives us the proportionality of the sides:

\[
\frac{PQ}{QS} = \frac{PT}{QT}
\]

Substituting the known values:

- \( PQ = RP = 7 \, \text{cm} \)
- \( QS = SQ + RP = 13 + 7 = 20 \, \text{cm} \)
- \( PT = 35 \, \text{cm} \)

Thus, the ratio becomes:

\[
\frac{7}{20} = \frac{35}{QT}
\]

Now, solve for \( QT \):

\[
QT = \frac{35 \times 20}{7} = 100 \, \text{cm}
\]

The length of \( QT \) is **100 cm**. However, the answer options do not include 100 cm, suggesting a review of the given lengths is necessary.

In the figure, PQ is parallel to RS. The length of RP is 7 cm; the length of PT is 35 cm; the length of SQ is 13 cm. What is the length of QT?

Solve the problem where PQ is parallel to RS, with given side lengths RP = 7 cm, PT = 35 cm, and SQ = 13 cm. Learn how to use the Basic Proportionality Theorem to find the length of QT using similar triangles.

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