# Height of the lamp post

Geometry Level 5

John is walking by the edge of a road at night.

There is a lamp post besides the road, which is working. The nearest point on the road to the lamp post is $$P$$.

His shadow is produced due to the light from the lamp post.

When he walks from point $$R$$ to point $$Q$$, he notices that the length of his shadow at point $$R$$ was $$\dfrac{3}{2}$$ times the length of his shadow at point $$Q$$.

Next, when John walks from point $$Q$$ to point $$P$$, he further notices two things.

$$1)$$ $$Q$$ is halfway between $$P$$ and $$R$$.

$$2)$$ The length of his shadow when he was at point $$Q$$ and the distance between him and the pole at point $$P$$ are equal.

On the basis of the data collected by John, find the height of the lamp post (in meters), if his height is $$1.8 \ m$$.

$$\textbf{Details And Assumptions}$$

$$\bullet$$ $$P$$, $$Q$$ and $$R$$ are co-linear and are on the edge of the road.

$$\bullet$$ Ignore any light other than that coming from the lamp post in question.

$$\bullet$$ Source : Problems plus in IIT Mathematics by Asit Das Gupta.

$$\bullet$$ Image Credit : Red Edge

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