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 $CF$ (in cm)

• A. 9 cm
• B. 8 cm
• C. 10 cm
• D. 12 cm

Answer 1

The correct option is B 8 cm

Given $AB=20cm,AD=12cm$
$DC=AB=20cm$ and $AD=BC=12cm$
Given that bisector of $\angle A$ intersects $DE$ 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)
$△APF\cong △ABF$

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