Question

Suppose $ABCD$ is a rhombus. A straight line passing through $C$ meet $AD$ which is produced at $P$ and meet $AB$ produced at $Q$. Therefore if $DP=\frac{1}{2}AB$, then find the ratio between $BQ$ and $AB$?

• A. $2:1$
• B. $1:1$
• C. $1:3$
• D. $3:1$

Answer 1

The correct option is A $2:1$

$ABCD$ is a rhombus. $AB=BC=CD=DA$

$\frac{DP}{AB}=\frac{1}{2}\Rightarrow DP=1AB=2$

In rhombus $\angle {\theta }_1=\angle {\theta }_2$

$\angle Q$ is common for the $△BCQand△APQ$

$\because \angle APQ=\angle BCQ$

$\because △BCQ$ is similar to $△APQ$ by $AAA$ property.

$\frac{AP}{BC}=\frac{AQ}{BQ}=\frac{AD+DP}{BC}=\frac{3}{2}$

$\frac{AQ}{BQ}=\frac{3}{2}\Rightarrow \frac{AB+BQ}{BQ}=\frac{3}{2}$

$\frac{AB}{BQ}=\frac{3}{2}-1=\frac{1}{2}$

$\frac{BQ}{AB}=\frac{2}{1}$