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A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60°, when he retires 40 meters from the bank then he find the angle to be 30°. Then the breadth of the river is ?
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- 40 m
- 60 m
- 20 m
- 30 m
Correct Option: C
Let us draw a figure below as per given question.
Let A be the position of a person on the bank of a river and OP the tree on the opposite bank and ∠OAP = 60°. When the person retires to the position B, then AB = 40 meter and ∠OBP = 30°
Let us assume OA(Breadth of the river) = x meter and height of tree OP = h meter
In ΔOAP, Use the trigonometry formula
Tan60° = P/B = Perpendicular distance / Base distance
⇒ Tan60° = OP / OA
⇒ OP = OA Tan60°
Put the value of OP and OA, We will get
⇒ h = x√3 ..............(1)
Now in the triangle ΔOBP
Tan30° = OP / OB
⇒ OP = OB Tan30°
⇒ OP = (x + 40)/√3
⇒ h = (x + 40)/√3 ...................(2)
From Equation (1) and (2), We will get
⇒ (x + 40)/√3 = x√3
⇒ (x + 40) = x√3 X √3
⇒ (x + 40) = 3x
⇒ 3x - x = 40
∴ x = 20 m
