Dear Reader, Below are three problems based on geometry.
Jaya Towers and Karuna towers are two tall buildings standing 400 metres apart. Two fast flying birds were sitting on top of these two towers and were aiming at catching the grains put up in one place. Jaya Tower’s height is 300 metres and the grains 160 metres apart from Jaya Towers. Both the birds were flying down at the same speed and reached the grains spot at the same time. What is the approximate height of Karuna Towers?
a) 240 metres b) 200 metres c) 180 metres d) none of these
Answer : a) 240 metres.
Jaya Tower is 300 metre tall -- Let Karuna Tower’s height be x
Distance of grains spot from Jaya Towers -- 160 metres.
Both the birds were flying down at the same speed and reached the grains spot at the same time
i.e Time taken for bird on Jaya tower to reach grain = Time taken for bird on Karuna tower to reach grain
Or Distance travelled by bird on Jaya tower / Speed of bird on Jaya tower = Distance travelled by bird on Karuna tower / Speed of bird on Karuna tower ->(1)
But as it is given that both the birds travelled with same speed, Speed of bird on Jaya tower = Speed of bird on Karuna tower. Applying this to equation (1) we get,
Distance travelled by bird on Jaya tower = Distance travelled by bird on Karuna tower -> (2)
Apply Pythagoras theorem, we know that (Distance travelled by bird on Jaya tower)2 = (300)2 + (160)2 -> (3)
Again by applying Pythagoras theorem to the other triangle. (Distance travelled by bird on Karuna tower)2 = (400-160)2 + x2 -> (4)
From equations 2,3 and 4 we get
(300)2 + (160)2 = (400-160)2 + x2
115600 = 57600 + x2
58000 = x2
240.83 m = x
Therefore x is approximately 240 m
Two famous temples, one on each bank of a river are there, opposite to each other. One temple Siva Temple -- is 54 m high. From the top of this temple, the angles of depression of the top and the foot of the other temple —Vishnu Temple are 30 deg. And 60 degrees respectively. Find the width of the river and the height of Vishnu Temple tower.
a)24.3m, 18 m b)31.14m ,36 m c)34.5m ,54 m d) none of these.
Answer : b)31.14m ,36 m
Let AB and CD be the two temples and AC be the river.
Then AB = 54 m
Let AC = x metres and CD = h metres
Angle ACB = 60 deg., angle EDB = 30 deg.
AB/AC = tan 60 deg. = √3
= > AC = AB/√3 = 54/√3 = [ 54/√3 x √3/√3] = 18 √3 m
DE = AC = 18 √3 m
BE / DE = tan 30 deg. = 1/ √3
BE = [ 18√3 x 1/ √3] = 18 m
Therefore CD = AE = AB - BE = (54 - 18) m = 36 m
So, width of the river = AC = 18 √3 m = (18 x 1.73) m = 31.14 m
Height of Vishnu temple = CD = 36 m
Satish was looking at a new building Kohli Towers --constructed having height of 200 m. The angle of elevation of the top of building from a point on ground is 30 deg. What is the distance of the point from the foot of the building?
a) 273 m b) 254 m c) 346 m d) 166 m
Answer : c) 346 m
Angle APB = 30 deg. And AB = 200 m
AB/AP = tan 30 deg. = 1/√3
AP = (AB x √3) m
200√3 m = 200 x 1.73