Question

In a parallelogram ABCD, AB = 20 cm and AD = 12 cm. The bisector of angle A meets DC at E and BC produced at F. Find the length of CF.

Answer 1

Given $AB=20cm,AD=12cm$
$DC=AB=20cm$ and $AD=BC=12cm$
Given that bisector of $\angle A$ intersects $DC$ at $E$ and $BC$ produced at $F$.
Draw $PF\parallel CD$
From the figure, $CD\parallel FP$ and $CF\parallel DP$
Hence $PDCF$ is a parallelogram.
$AB\parallel FP$ and $AP\parallel BF$
$ABFP$ is also a parallelogram
Consider $△APF$ and $△ABF$
$\angle APF=\angle ABF$ [Since opposite angles of a parallelogram are equal]
$AF=AF$ (Common side)
$\angle PAF=\angle AFB$ (Alternate angles)
$△PAF\cong △BFA$ (By ASA congruence criterion)

$AB=AP$
$AB=AD+DP$
$=AD+CF$ [Since $DCFP$ is a parallelogram]
$CF=AB-AD$
$=(20-12)cm=8cm$