In the picture,the vertical lines are equally spaced prove that their heights are in the arithmetic sequence. What is the common difference?
Answers
Given : vertical lines are equally spaced
To find : Prove that, their heights are in the arithmetic sequence
Solution:
Let say Horizontal Distance of First Perpendicular = a
and then Equally Spaced at d
Hence Distances are
a , a + d , a + 2d , ................., a + (n-1)d
p₁ , p₂ , p₃ .....................pₙ are height
Lets compare two triangle one with base a and another with a + d
Now one angle is common
& another is 90°
Hence Similar Triangle
=> (a + d)/a = p₂/p₁
=> p₂/p₁ = 1 + d/a
=> p₂ = p₁ + p₁ (d/a)
=> p₂ = p₁ + (p₁d/a)
Similarly other triangles are Similar
(a + 2d)/a = p₃/p₁
=> 1 + 2d/a = p₃/p₁
=> p₃ = p₁ + (p₁/a)2d
=> p₃ = p₁ + 2(p₁d/a)
(a + (n-1)d)/a = pₙ/p₁
=> 1 + (n-1)d/a = pₙ/p₁
=> pₙ = p₁ + (n-1)(p₁d/a)
p₁ , p₁ + (p₁d/a) , p₁ + 2(p₁d/a) , .................... p₁ + (n-1)(p₁d/a)
This is an AP
Where first term = p₁
& common difference = p₁d/a = d.tan40° ( as p₁/a = tan40° )
Hence proved height form an arithmetic sequence.
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Answer:
Step-by-step explanation:
Given : vertical lines are equally spaced
To find : Prove that, their heights are in the arithmetic sequence
Solution:
Let say Horizontal Distance of First Perpendicular = a
and then Equally Spaced at d
Hence Distances are
a , a + d , a + 2d , ................., a + (n-1)d
p₁ , p₂ , p₃ .....................pₙ are height
Lets compare two triangle one with base a and another with a + d
Now one angle is common
& another is 90°
Hence Similar Triangle
=> (a + d)/a = p₂/p₁
=> p₂/p₁ = 1 + d/a
=> p₂ = p₁ + p₁ (d/a)
=> p₂ = p₁ + (p₁d/a)
Similarly other triangles are Similar
(a + 2d)/a = p₃/p₁
=> 1 + 2d/a = p₃/p₁
=> p₃ = p₁ + (p₁/a)2d
=> p₃ = p₁ + 2(p₁d/a)
(a + (n-1)d)/a = pₙ/p₁
=> 1 + (n-1)d/a = pₙ/p₁
=> pₙ = p₁ + (n-1)(p₁d/a)
p₁ , p₁ + (p₁d/a) , p₁ + 2(p₁d/a) , .................... p₁ + (n-1)(p₁d/a)
This is an AP
Where first term = p₁
& common difference = p₁d/a = d.tan40° ( as p₁/a = tan40° )
Hence proved height form an arithmetic sequence.
Hope you understand
Thank you