Question

Given the measurements in the following figure : If $AD=m\sec \theta$ or $12cosec\theta$
Then, the value of $m$ is :

• A. $7$
• B. $3$
• C. $2$
• D. $9$

Answer 1

Answer: (D)

$9$

Draw a perpendicular from $D$ on $AB$ to meet at $AB$ at $E$. Hence, $DEBC$ is a rectangle.
Thus, $DE=BC=12$
In $△DBC$,
$B{D}^2=C{D}^2+B{C}^2$
${13}^2={12}^2+C{D}^2$
$C{D}^2=25$
$CD=5$
Now, $BE=CD=5$
Thus, $AE=AB-BE$
$AE=14-5$
$AE=9$
Now, In $△AED$
$\sin \theta =\frac{DE}{AD}=\frac{12}{AD}$
$AD=12cosec\theta$
$\cos \Theta =\frac{AE}{AD}=\frac{9}{AD}$
$AD=9\sec \theta$