Question

ABDE is a trapezium​​​​​​ with $AE\parallel BD$ and $ED\perp BD$, and AC is drawn parallel to the side DE.

Find the values of $\text{cot}\theta and\text{cosec}\theta$ respectively, from the given figure.

• A. $\frac{5}{4},\frac{5}{4}$
• B. $2,\frac{5}{2}$
• C. $\frac{4}{3},\frac{5}{3}$
• D. $\frac{5}{2},2$

Answer 1

The correct option is C

$\frac{4}{3},\frac{5}{3}$

Consider the figure ACDE
$AE\parallel CD$, $AC\parallel DE$ and $ED\perp CD$

$\therefore$ ACDE is a rectangle

CD = AE = 10 units

BC = BD - CD = 16 - 10 = 6

Now, in $△ABC$,

$\angle C={90}^{\circ }$

Now, from pythagoras theorem,

$A{C}^2+B{C}^2=A{B}^2$

$A{C}^2+6^2={10}^2$

$AC=\sqrt{100-36}=\sqrt{64}=8$

Thus,
$\text{cot}\theta =\frac{AC}{BC}=\frac{8}{6}=\frac{4}{3}$
$\text{and}\text{cosec}\theta =\frac{AB}{BC}=\frac{10}{6}=\frac{5}{3}$