A 1.5 m tall person is standing at a distance of 3 m from a lamp post. The light from the lamp at the top of the post casts her shadow. The length of the shadow is twice her height. What is the height of the lamp post in meters?
1.5
3
4.5
6
ANSWER
The lamp post is 3 m tall, so the answer is option (B).
Official answer: BSource: Official answer key, GATE 2018 EC (IIT Guwahati)
Length of the shadow = twice her height: . The shadow falls on the side of the person away from the lamp post.
So the tip of the shadow is from the foot of the lamp post.
The light ray from the lamp passes over the top of her head and ends at the tip of the shadow. This gives two right triangles that share the angle at the shadow's tip: the lamp post (height ) with base , and the person (height ) with base . Both stand vertically, so the triangles are similar.
In similar triangles the ratio of height to base is the same:
Therefore , which is option (B).
KEY IDEA
The lamp, the top of the person's head and the tip of the shadow lie on one straight light ray. The lamp post and the person are both vertical, so they form two similar right triangles that share the angle at the shadow's tip. Measure each base from the tip of the shadow: height / base is the same for both triangles.
WATCH OUT FOR
Using the 3 m person-to-post distance as the lamp post's base and leaving out the shadow. That gives H/3 = 1.5/3, so H = 1.5 m (option A). The lamp post's base has to run from its foot to the tip of the shadow, 3 + 3 = 6 m. A second slip is taking the shadow as 1.5 m instead of twice her height (3 m).
From the GATE 2018 EC paper. Every question of this paper · Download the full paper · All 22 Geometry questions, with notes
