Standing on a river bank, a person observes the top of a tower on the opposite bank at 45° elevation. Which statement is correct about the river’s breadth compared to the tower’s height?
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ABreadth of the river is half of the height of the tower
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BBreadth of the river and the height of the tower are equal
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CBreadth of the river is twice the height of the tower
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DNone of these
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EBreadth is thrice the height
Answer
Correct Answer: Breadth of the river and the height of the tower are equal
Explanation
Introduction / Context:The 45° angle of elevation to the top of a vertical object on level ground is a signature configuration: the opposite and adjacent legs of the right triangle are equal.
Given Data / Assumptions:
- Angle of elevation θ = 45°.
- Ground is level; tower is vertical.
Concept / Approach:tan 45° = 1 ⇒ opposite/adjacent = 1 ⇒ height = horizontal distance. Here, the horizontal distance equals the river’s breadth.
Step-by-Step Solution:
Let height = H and breadth = B. tan 45° = H/B = 1 ⇒ H = B.Verification / Alternative check:Any angle greater than 45° would imply H > B; any angle less than 45° implies H < B. With 45°, equality holds.
Why Other Options Are Wrong:“Half” or “twice” contradicts tan 45° = 1; “None of these” is false because equality is exactly true.
Common Pitfalls:Confusing sine/cosine with tangent; overlooking that the breadth equals the adjacent side in the triangle.
Final Answer:Breadth of the river and the height of the tower are equal