Question

In the given figure, ABCD is a quadrilateral such that $DA\perp AB$ and $CB\perp AB$, x = y and EF || AD such that EF bisects $\angle DEC$, which is equal to 90 degrees. Then $C{D}^2$ is equal to

• A. $2C{E}^2$
• B. $2D{E}^2$
• C. Both (a) and (b)
• D. None of these

Answer 1

The correct option is C

Both (a) and (b)

Given,
ABCD is a quadrilateral with $DA\perp AB$ and $CB\perp AB$
and EF || AD
$\Rightarrow EF||BC$
$\Rightarrow EF\perp AB$

Given that

EF bisects $\angle DEC$

Therefore, $\angle DEF=\angle CEF$

Also, AD is parallel to EF.

Therefore, $\angle ADE=\angle DEF$

(Since they are alternate interior angles)

Also, EF is parallel to BC.

Therefore, $\angle BCE=\angle CEF$

(Since they are alternate interior angles)

Therefore, $\angle BCE=\angle ADE$

(Since $\angle DEF=\angle CEF$)

Now, in $\Delta DAE$ and $\Delta CBE$,
$\angle DAE=\angle CBE$ [each ${90}^{\circ }$]
Also, $\angle ADE=\angle BCE$
$\therefore \Delta DAE\sim \Delta CBE$ [by AA similarity]
$\Rightarrow \frac{DE}{AE}=\frac{CE}{BE}\Rightarrow DE=CE$ ...(i)
Now, in $\Delta DEC$, right angled at E,
$D{E}^2+C{E}^2=D{C}^2\Rightarrow 2D{E}^2=2C{E}^2=D{C}^2$