Question

$AB,$ if $DC=20cm$ and $MN=27cm$.

• A. $43cm$
• B. $31cm$
• C. $34cm$
• D. $30cm$

Answer 1

The correct option is C $34cm$

Join $BD$

Let $BD$ and $MN$ meet at $Q$.

Since, $M$ is the mid point of $AD$ and $N$ is the mid point of $BC$.

So by mid-point theorem, $AB\parallel MN\parallel CD$
In $△BDC$ and $△BQN$
$\angle B=\angle B$ (Common)
$\angle BDC=\angle BQN$ ..... (Corresponding angles of parallel lines)
$\angle BCD=\angle BNQ$ ..... (Corresponding angles of parallel lines)
Thus, $△BDC\sim △BQN$
Thus, $\frac{BD}{QB}=\frac{DC}{QN}$
$2=\frac{DC}{QN}$ ..... ($Q$ is the mid-point of $BD$)
$QN=\frac{1}{2}DC$
Similarly, $QM=\frac{1}{2}AB$
Hence, $QM+QN=\frac{1}{2}(AB+DC)$
$MN=\frac{1}{2}(AB+CD)$
Therefore, $2\times 27=AB+20$
$\Rightarrow AB=54-20=34cm$