Similarity — Case-Study Based 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 :

(i) What is the height of the tower ?

(ii) What is Rohan's height ?

(iii) What will be the length of the shadow of the lamp post when the tower casts a shadow of 35 m ?

Answer

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,

△ABC ∼ △DEF

We know that,

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

$\Rightarrow \dfrac{AB}{BC} = \dfrac{DE}{EF} \\[1em] \Rightarrow \dfrac{5}{4} = \dfrac{DE}{28} \\[1em] \Rightarrow DE = \dfrac{5}{4} \times 28 = 5 \times 7 = 35 \text{ m}.$

Hence, the height of the tower = 35 m.

(ii) Thus,

△ABC ∼ △PQR

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

$\Rightarrow \dfrac{AB}{BC} = \dfrac{PQ}{QR} \\[1em] \Rightarrow \dfrac{5}{4} = \dfrac{PQ}{1} \\[1em] \Rightarrow PQ = \dfrac{5}{4} \times 1 = \dfrac{5}{4} = 1.25 \text{ m}.$

Hence, Rohan's height = 1.25 m.

(iii) When tower casts a shadow of 35 m,

Due to similarity,

$\Rightarrow \dfrac{\text{Height of tower}}{\text{Length of shadow of tower}} = \dfrac{\text{Height of lamp}}{\text{Length of shadow of lamp}} \\[1em] \Rightarrow \dfrac{35}{35} = \dfrac{5}{\text{Length of shadow of lamp}} \\[1em] \Rightarrow 1 = \dfrac{5}{\text{Length of shadow of lamp}} \\[1em] \Rightarrow \text{Length of shadow of lamp} = 5 \text{ m}.$

Hence, length of shadow of lamp = 5 m.