Question

The length of the ladder placed against a vertical wall is twice the distance between the foot of the ladder and the wall.

Assertion (A): The angle that the ladder makes with the wall is 60°

Reason (R): If ladder makes angle θ with the wall then \[\sin \theta=\frac{\text { perpendicular }}{\text { hypotenuse }}=\frac{x}{2 x}=\frac{1}{2}\]

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A .

Answer 1

A is false, R is true.

Explanation:

Let AB be the wall and AC be the ladder. Let distance between foot of ladder and wall be x.

So, length of ladder (AC) = 2x.

According to the Pythagoras theorem:

\[\Rightarrow \text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2\]

\[\Rightarrow AC^2 = BC^2 + AB^2\]

\[\Rightarrow (2x)^2 = x^2 + AB^2\]

\[\Rightarrow 4x^2 = x^2 + AB^2\]

\[\Rightarrow AB^2 = 4x^2 - x^2\]

\[\Rightarrow AB^2 = 3x^2\]

\[\Rightarrow AB = \sqrt{3x^2}\]

\[\Rightarrow AB = \sqrt{3}\,x.\]

Let angle between ladder and wall be θ.

We know that,

\[\Rightarrow \sin \theta = \frac{\text{Perpendicular side}}{\text{Hypotenuse}}\]

\[\Rightarrow \sin \theta = \frac{BC}{AC}\]

\[\Rightarrow \sin \theta = \frac{x}{2x}\]

\[\Rightarrow \sin \theta = \frac{1}{2}\]

\[\Rightarrow \sin \theta = \sin 30^\circ\]

\[\Rightarrow \theta = 30^\circ.\]

Assertion (A) is false.

If ladder makes angle θ with the wall then \[\sin \theta = \frac{\text{base}}{\text{hypotenuse}} = \frac{x}{2x} = \frac{1}{2}.\]

Here base refers side opposite to angle θ i.e. BC.

Reason (R) is true.