Question

A ladder leans a wall making an angle of ${60^ \circ }$with the ground. The foot of the ladder is $2m$away from the wall. Find the length of the ladder.
A) $8m$
B) $6m$
C) $4m$
D) $2m$

Answer 1

Hint: We can solve this question using trigonometry. Since the wall and the ground will be perpendicular, we can consider a right-angled triangle here. Also, one of the non-right angles is given as ${60^ \circ }$. So, using these values and trigonometric relations we can find the answer.

Formula used:
For a right-angled triangle with one of the non-right angles $\theta $,
$\cos \theta = \dfrac{{{\text{adjacent side}}}}{{{\text{Hypotenuse}}}}$

Complete step-by-step answer:
Given that a ladder leans a wall making an angle of ${60^ \circ }$with the ground and the foot of the ladder is $2m$away from the wall.
We can construct a figure according to the given information.
Since the ground and wall is always perpendicular to each other, we get a right-angled triangle.
Also, the angle made by the ladder in the ground is given as ${60^ \circ }$.
Noting that the distance between the foot of the ladder and the wall is $2m$, we get the following figure.

Now we are asked to find the length of the ladder. We can see from the figure that the ladder represents the hypotenuse of the triangle and the $2m$represents the adjacent side of the ${60^ \circ }$ angle.
For a right-angled triangle with one of the non-right angles $\theta $,
$\cos \theta = \dfrac{{{\text{adjacent side}}}}{{{\text{Hypotenuse}}}}$
So, we can use this result here.
Here, $\theta = {60^ \circ }\& {\text{adjacent side = 2m}}$
So, we have,
$\Rightarrow$ $\cos {60^ \circ } = \dfrac{{\text{2}}}{{{\text{Hypotenuse}}}}$
We know $\cos {60^ \circ } = \dfrac{1}{2}$.
Substituting this we get,
$\Rightarrow$ $\dfrac{1}{2} = \dfrac{{\text{2}}}{{{\text{Hypotenuse}}}}$
Cross multiplying, we get,
$\Rightarrow$ ${\text{Hypotenuse}} = 2 \times 2 = 4$
That hypotenuse of the triangle is $4m$.
But we had seen that this hypotenuse is equal to the length of the ladder.
So, the length of the ladder is $4m$.
$\therefore $ The answer is option C.

Note: Using this right triangle and trigonometric relations we can also find the height of the wall where the ladder meets the wall also.
If, instead of this angle we were given the other non-right angle we can either find this angle by subtracting the given angle from ${90^ \circ }$. Or we can use another formula.
$\sin \theta = \dfrac{{{\text{Opposite side}}}}{{{\text{Hypotenuse}}}}$