Angles in progression

Find the type of progression formed by the angle between the tangents to a circle drawn from points which are collinear and at equal distances from each other as depicted.

#Geometry #Sequences #TangentOfCircles

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

All I got is that cosec of half of angles are in AP.

Krishna Sharma - 5 years, 11 months ago

That's interesting. Will you please let me know about the working part?

Rohit Ner - 5 years, 11 months ago

I am not sure that you wanted what I have written above but it was pretty easy to get

Draw a line passing through centre and all the points
Let the radius of circle be $R$ , distance between centre and first point be $d$ and distance between consecutive points be $x$
We will take triangle formed by centre, point of contact and the first point(right angled triangle)
Let the original first angle be $\theta$ then the angle of triangle be $\theta/2$

And we will get $sin(\theta/2) = \dfrac{R}{d}$ .

Similarly for next angle $sin(\theta_1/2) = \dfrac{R}{d+x}$

And for next $sin(\theta_2/2) = \dfrac{R}{d + 2x}$

Now rest is easy just flip them to make cosec and eliminate R,d,x to get desired result :)

Krishna Sharma - 5 years, 11 months ago

@Krishna Sharma – Marvelous! Thank you very much for your guiding. I would try to build upon your work. :)

Rohit Ner - 5 years, 11 months ago

You should mention what the curve is. It doesn't quite seem like a circle to me, but that might be because of the straight lines.

Calvin Lin Staff - 5 years, 11 months ago

Done! :)

Rohit Ner - 5 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$