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 PP.

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

When he walks from point RR to point QQ, he notices that the length of his shadow at point RR was 32\dfrac{3}{2} times the length of his shadow at point QQ.

Next, when John walks from point QQ to point PP, he further notices two things.

1) 1) QQ is halfway between PP and RR.

2) 2) The length of his shadow when he was at point QQ and the distance between him and the pole at point PP 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 m1.8 \ m.

Details And Assumptions\textbf{Details And Assumptions}

\bullet PP, QQ and RR 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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