Perpendicular Chords of a Circle

Help Required: Given a point P in the interior of a circle of radius 1 unit with two perpendicular chords through P and another pair of perpendicular chords through P that make an angle of θ radians with the first pair, as shown in the figure, find the area of the shaded region.

Easy Math Editor

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`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

Hint: As suggested by the problem, the answer is independent of the point (as long as it's contained within).

Now, figure out how to prove this fact.

In particular, what can we say about how the circumference is split into these regions?

Calvin Lin Staff - 5 years, 5 months ago

We can consider specific cases and conclude that the answer is 2θ but so far I have not been able to prove it for any general point within the circle. Would 2θ be correct?

Ajit Athle - 5 years, 5 months ago

Yes, that conjecture is correct.

Calvin Lin Staff - 5 years, 5 months ago

I don't understand can you help me?

Department 8 - 5 years, 5 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}`	$\boxed{123}$