ABCD is a parallelogram and M is the mid point of BC. AB and DM are produced to meet at N, then :

Options

A. A)`AN=sqrt3AB`
B. B)`AN=2AB`
C. C)`AN^(2)=(3)/(2)AB^(2)`
D. D)`AN^(2)=2AB^(2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Parallelogram
We have a parallelogram ABCD, where opposite sides are parallel and equal in length. This means:
- AB is parallel to CD
- AD is parallel to BC

### Step 2: Identify the Midpoint
M is the midpoint of side BC. Therefore, we can say that:
- BM = MC

### Step 3: Extend the Lines
We extend lines AB and DM to meet at point N.

### Step 4: Apply Thales' Theorem
Since M is the midpoint of BC, we can apply Thales' theorem. According to this theorem, if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

In our case:
- Since AD is parallel to BC and M is the midpoint of BC, we can say that:
\[ \frac{AN}{AB} = \frac{AM}{BM} \]

### Step 5: Establish Ratios
Since M is the midpoint of BC, we have:
- BM = MC, which means AM is equal to BM (because M divides BC into two equal parts).

Thus, we can say:
\[ \frac{AM}{BM} = 1 \]

### Step 6: Relate AN and AB
From the ratios established, we can write:
\[ \frac{AN}{AB} = 2 \quad \text{(because AN = AB + BN and BN = AB)} \]
This implies:
\[ AN = 2 \times AB \]

### Conclusion
Thus, the relation between AN and AB is:
\[ AN = 2 \times AB \]

### Final Answer
The correct option is B: \( AN = 2 \times AB \).

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