In triangle ABC, M is mid-point of AB and a straight line through M and parallel to BC cuts AC at N. Find the lengths of AN and MN, if BC = 7 cm and AC = 5 cm.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Geometry of the Triangle
In triangle ABC, M is the midpoint of side AB. A line through M is drawn parallel to side BC, which intersects side AC at point N.

### Step 2: Identify the Given Lengths
We are given:
- Length of BC = 7 cm
- Length of AC = 5 cm

### Step 3: Use the Midpoint Theorem
According to the Midpoint Theorem, if a line segment is drawn parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Since M is the midpoint of AB, and the line through M is parallel to BC, point N will also be the midpoint of AC.

### Step 4: Calculate Lengths
1. **Finding MN**: Since MN is parallel to BC and M is the midpoint of AB, MN will be half the length of BC.
\[ MN = \frac{1}{2} \times BC = \frac{1}{2} \times 7 \text{ cm} = 3.5 \text{ cm} \]

2. **Finding AN**: Since N is the midpoint of AC, AN will be half the length of AC.
\[ AN = \frac{1}{2} \times AC = \frac{1}{2} \times 5 \text{ cm} = 2.5 \text{ cm} \]

### Final Results
- Length of MN = 3.5 cm
- Length of AN = 2.5 cm

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