if two trees of height x and y is standing on the two ends of a road subtend angles of 30 and 60 degree respectively at the midpoint of the road, then the ratio x:y is
Answers
If the two trees subtend 30 and 60-degree angles at the midpoint of the road then the ratio of heights x:y is 1:3.
Step-by-step explanation:
Referring to the figure attached below, we have
- x = height of the tree that subtends 30° angle at point C
- y = height of the tree that subtends 60° angle at point C
- Point C is the midpoint of the road AB ∴ AC = BC …… (i)
Consider ∆ ACD, applying the trigonometry ratio of a right triangle, we get
tan θ = perpendicular/base = DA/AC
⇒ tan 30° = x/AC
⇒ 1/√3 = x/AC
⇒ x = AC/√3 ….. (ii)
Consider ∆ BCE, applying the trigonometry ratio of a right triangle, we get
tan θ = perpendicular/base = EB/BC
⇒ tan 60° = y/BC
⇒ √3 = y/BC
⇒ y = √3 BC ….. (iii)
Thus, diving the eq. (ii) & (iii), we get
The ratio x:y as,
= [AC/√3] / [√3 BC]
= [AC] / [3 BC]
Since from (i) AC = BC
= 1/3
= 1 : 3
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