Question

If PQRS is a parallelogram, with point A on QR, then $\frac{AP}{AB}$ is equal to

• A. 1
• B. 2
• C. $\frac{1}{2}$
• D. can't be determined

Answer 1

The correct option is A

1

In $△$BPS,

RA || SP (PQRS is a parallelogram)

$\frac{BR}{RS}$ = $\frac{AB}{AP}$ [By BPT]

$\frac{x}{x}$ = $\frac{AB}{AP}$

$\frac{AB}{AP}$ = 1

Alternatively,

Given, PQRS is a parallelogram.
$\Rightarrow$ PQ || RS
or PQ || BS
$\Rightarrow \angle APQ=\angle ABR$ [alternate angles]
$\angle PAQ=\angle BAR$ [vertically opposite angles]
$\Rightarrow \Delta APQ\sim \Delta ABR$ [by AA similarity]
$\Rightarrow \frac{AP}{AB}=\frac{PQ}{BR}$
$\Rightarrow \frac{AP}{AB}=\frac{x}{x}$ [as PQ = SR =x]
$\Rightarrow \frac{AP}{AB}=1$