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How do I find the point $p$ where the arch meets the red line if the angle of the blue are is known and the height of the yellow?

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2 Answers 2

If $P(h,k),$ its projection on the bold black line will be $(h,0)$

The the height of the yellow is $k$(known)

The length of the projection on the bold black line will be $h$

So, the tangent of the known angle blue $=\frac kh\implies h=k\cdot$(the cotangent of the angle blue)

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Well the $y$ coordinate is obviously the height of the yellow line, so that is something we know.

For the $x$ coordinate we can use the tangent. Let $\alpha$ be your angle. It holds that $\tan(\alpha)= \frac{\text{height of the yellow line}}{\text{distance from the center to the yellow line in x direction}}$

As "distance from the center to the yellow line in x direction" equals the $x$ coordinate of P we have

$x = \frac{1}{\tan(\alpha)}\cdot \text{height of the yellow line}$

So P has the coordinates P($\frac{1}{\tan(\alpha)}\cdot \text{height of the yellow line}, \text{height of the yellow line}$)

All this assumes, that the middle of the circle has the coordinates (0,0). Otherwise just shift the result.

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