Question

ABCD is a quadrilateral with right angles at A and C. Points E and F are on the diagonal AC such that DE and BF are both perpendicular to AC. If AE $=3$ cm, DE $=5$ cm and CE $=7$ cm then the length of BF is ______ cm.

Answer 1

In $\Delta ADE$,

$\angle EAD+\angle BAF={90}^o$ and

$\angle EAD+\angle BAF={90}^o\Rightarrow \angle EDA=\angle BAF$

$\Rightarrow \frac{DE}{AF}=\frac{AE}{BF}$

$\Rightarrow \frac{AF}{BF}=\frac{DE}{AE}=\frac{5}{3}$....(1)

similarly $\Delta CFB:\Delta DEC$

$\Rightarrow \frac{CE}{BF}=\frac{DE}{CF}$

$\Rightarrow \frac{CF}{BF}=\frac{DE}{CE}=\frac{5}{7}$...(2)

adding (1) and (2)

$\frac{AF+CF}{BF}=\frac{50}{21}$

$\Rightarrow \frac{AC}{BF}=\frac{50}{21}$

$\Rightarrow \frac{10}{BF}=\frac{50}{21}$

$\Rightarrow BF=4.2$