A ladder rests against a vertical wall and makes an angle \(\theta\) with the horizontal ground such that \(\tan\theta = \dfrac{5}{12}\). If the length of the ladder is 13 meters, find the height at which the ladder touches the wall.
5 meters
The ladder, the wall and the ground form a right-angled triangle, with the ladder as the hypotenuse of length 13 m.
Given \(\tan\theta = \dfrac{5}{12} = \dfrac{\text{opposite}}{\text{adjacent}}\), the height and base are in the ratio 5 : 12.
The height at which the ladder touches the wall is the side opposite \(\theta\), so we use the sine ratio.
For a 5 : 12 right triangle the hypotenuse is \(\sqrt{5^2 + 12^2} = \sqrt{169} = 13\), hence \(\sin\theta = \dfrac{5}{13}\).
Height \(= \text{ladder} \times \sin\theta = 13 \times \dfrac{5}{13}\).
\(= 5\) meters.
So the ladder touches the wall at a height of 5 meters.
Hence the required height is 5 meters.
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