Can you find the radius?

Assuming you already know 'a', is it possible to find the value of the radius? How would you do it?

#Geometry

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`2 \times 3`	$2 \times 3$
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Comments

Assume that $x$ represents the arc length of a quarter of a circle.

Draw a non-vertical straight line from the center of the circle to the end-point of the arc length that represents the length of $a$ .

Define $\theta$ as the acute angle from this non-vertical straight line and the horizontal line shown.

Use the arc length formula, $s = r \Theta$ , we get $a = r \left( \frac \pi2 - \theta\right) \Rightarrow \theta = \frac \pi2 - \frac ar$ , where $r$ denotes the radius of the circle.

And by definition, $\cos \theta = \frac{2-r}{r} \Rightarrow r \cdot \underbrace{ \cos\left( \frac \pi2 - \frac ar \right)}_{=\sin\left(\frac ar\right)} = 2 -r$ . We are left with a transcendetal (singular variable) equation of $r$ .

Pi Han Goh - 2 years 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}`	$\boxed{123}$