If PQRS is a parallelogram, with point A on QR, then find the value of $\frac{A P}{A B}$ .

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Solution

In $△$ BPS,

RA || SP (PQRS is a parallelogram)

$\frac{B R}{R S}$ = $\frac{A B}{A P}$ [By BPT]

$\frac{x}{x}$ = $\frac{A B}{A P}$

$\frac{A B}{A P}$ = 1

Alternatively,

Given, PQRS is a parallelogram.
$\Rightarrow$ PQ || RS
or PQ || BS
$\Rightarrow ∠ A P Q = ∠ A B R$ [alternate angles]
$∠ P A Q = ∠ B A R$ [vertically opposite angles]
$\Rightarrow Δ A P Q \sim Δ A B R$ [by AA similarity]
$\Rightarrow \frac{A P}{A B} = \frac{P Q}{B R}$
$\Rightarrow \frac{A P}{A B} = \frac{x}{x}$ [as PQ = SR =x]
$\Rightarrow \frac{A P}{A B} = 1$

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