Question

Rohan is trying to find the height of a tower shown below. He is using the properties of similar triangles. He observes a lamp post near it of height 5 m casting a shadow of 4 m on the ground. At the same time, he himself is casting a shadow of 1 m on the ground.

Based on the above information, answer the following questions:

1. What is the height of the tower?
2. What is Rohan’s height?
3. What will be the length of the shadow of the lamp post when the tower casts a shadow of 35 m?

Answer 1

Since the lamp post, tower, and Rohan are all vertical to the ground and sunlight makes the same angle, the triangles formed are similar.

(i) Thus,

\[\triangle ABC \sim \triangle DEF\]

We know that,

In similar triangles, corresponding sides are proportional (in the same ratio).

⇒ \[\frac{AB}{BC} = \frac{DE}{EF}\]

⇒ \[\frac{5}{4} = \frac{DE}{28}\]

⇒ \[DE = \frac{5}{4} \times 28 = 5 \times 7 = 35\ \text{m.}\]

Hence, the height of the tower = \[35\ \text{m.}\]

(ii) Thus,

\[\triangle ABC \sim \triangle PQR\]

In similar triangles, corresponding sides are proportional (in the same ratio).

⇒ \[\frac{AB}{BC} = \frac{PQ}{QR}\]

⇒ \[\frac{5}{4} = \frac{PQ}{1}\]

⇒ \[PQ = \frac{5}{4} \times 1 = \frac{5}{4} = 1.25\ \text{m.}\]

Hence, Rohan's height = \[1.25\ \text{m.}\]

(iii) When tower casts a shadow of \[35\ \text{m}\],

Due to similarity,

⇒ \[\frac{\text{Height of tower}}{\text{Length of shadow of tower}} = \frac{\text{Height of lamp}}{\text{Length of shadow of lamp}}\]

⇒ \[\frac{35}{35} = \frac{5}{\text{Length of shadow of lamp}}\]

⇒ \[1 = \frac{5}{\text{Length of shadow of lamp}}\]

⇒ \[\text{Length of shadow of lamp} = 5\ \text{m.}\]

Hence, length of shadow of lamp = \[5\ \text{m.}\]