PQR is an equilateral triangle. MN is drawn parallel to QR such that M is on PQ and N is on PR. If PN = 6 cm, then the length of MN is

Options

A. 3 cm
B. 6 cm
C. 12 cm
D. 4.5 cm

Answer

To solve the problem, we will use the properties of similar triangles.

### Step-by-Step Solution:

1. **Identify the Triangle**: We have an equilateral triangle PQR. This means all sides are equal, and all angles are 60 degrees.

2. **Draw the Triangle**: Let's draw triangle PQR with points P, Q, and R. Mark the points M on PQ and N on PR such that MN is parallel to QR.

3. **Understand the Parallel Lines**: Since MN is parallel to QR, triangles PMN and PQR are similar by the Basic Proportionality Theorem (also known as Thales' theorem). This means that the ratios of corresponding sides of these triangles are equal.

4. **Set Up the Ratio**: Let the length of MN be x. Since MN is parallel to QR, we can set up the following ratio:
\[ \frac{PN}{PR} = \frac{MN}{QR} \]
Given that PN = 6 cm, and since PQR is equilateral, PR = QR.

5. **Express PR and QR**: Let the length of each side of the equilateral triangle PQR be 's'. Therefore, PR = QR = s.

6. **Substitute Known Values**: Substitute the known values into the ratio:
\[ \frac{6}{s} = \frac{x}{s} \]

7. **Solve for x**: Since the sides PR and QR are equal, we can simplify the equation:
\[ 6 = x \]
Therefore, the length of MN is 6 cm.

### Final Answer:
The length of MN is **6 cm**.

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