Similarity of Triangles — Case-Study Based Questions

Case study: Amit is trying to find the average height of a tower near his house. The height of Amit's house is 20 m. At a certain time of the day, when Amit's house casts a shadow 10 m long on the ground, the tower casts a shadow 50 m long on the ground and Sumit's house casts a 20 m long shadow on the ground.

Read the following case study carefully and answer the questions based on it:

What is the height of the tower?

(a) 50 m
(b) 80 m
(c) 100 m
(d) 150 m

Answer

At the given time, all objects and their shadows form similar right triangles, so the ratio of height to shadow is the same for every object.

For Amit's house : $\dfrac{\text{height}}{\text{shadow}} = \dfrac{20}{10} = 2$.

So, height of tower = 2 × (its shadow) = 2 × 50 = 100 m.

Hence, option (c) is the correct option.

What will be the length of the shadow of the tower when Amit's house casts a shadow 12 m long?

(a) 50 m
(b) 60 m
(c) 75 m
(d) 90 m

Answer

When Amit's house (20 m) casts a shadow of 12 m, the common ratio of height to shadow at that time is

$\dfrac{\text{height}}{\text{shadow}} = \dfrac{20}{12} = \dfrac{5}{3}.$

For the tower (height 100 m), shadow = $\dfrac{\text{height}}{\text{ratio}} = \dfrac{100}{\frac{5}{3}} = 100 \times \dfrac{3}{5} = 60$ m.

Hence, option (b) is the correct option.

What is the height of Sumit's house?

(a) 20 m
(b) 30 m
(c) 40 m
(d) 50 m

Answer

At the time when Amit's house casts a shadow of 10 m, the ratio of height to shadow is 2 (from Question 1), and Sumit's house casts a shadow of 20 m.

So, height of Sumit's house = 2 × 20 = 40 m.

Hence, option (c) is the correct option.

At a time when the tower casts a shadow 40 m long, what will be the length of the shadow of Sumit's house?

(a) 8 m
(b) 16 m
(c) 20 m
(d) 32 m

Answer

When the tower (height 100 m) casts a shadow of 40 m, the common ratio of height to shadow at that time is

$\dfrac{\text{height}}{\text{shadow}} = \dfrac{100}{40} = \dfrac{5}{2}.$

For Sumit's house (height 40 m), shadow = $\dfrac{40}{\frac{5}{2}} = 40 \times \dfrac{2}{5} = 16$ m.

Hence, option (b) is the correct option.

When the tower casts a shadow 30 m long, at the same time what will be the length of the shadow of Amit's house?

(a) 15 m
(b) 8 m
(c) 6 m
(d) 4 m

Answer

When the tower (height 100 m) casts a shadow of 30 m, the common ratio of height to shadow at that time is

$\dfrac{\text{height}}{\text{shadow}} = \dfrac{100}{30} = \dfrac{10}{3}$

For Amit's house (height 20 m), shadow = $\dfrac{20}{\frac{10}{3}} = 20 \times \dfrac{3}{10} = 6$ m.

Hence, option (c) is the correct option.