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If a line with slope a (in the plane) is reflected through another line with slope b, then the image line has slope c,
where c = (a*b^2-a+2*b)/(-b^2+1+2*b*a).
In this GeoGebra worksheet, the blue line is the reflected image of the red line through the black line. Adjust the sliders to change the slopes of the lines.

This applet demonstrates the standard ruler and compass construction for the square root of a number. Drag the point C to see how it works.

The equation sin(x) = c has two solutions in the interval [0°, 360°) when -1 < c < 1. If c = 1 or c = -1 then there is only one solution in this interval.
Drag the slider to visualize these solutions on the unit circle.

You are walking from one end of an airport terminal to the other. The airport has several moving walkways, and you need to stop to tie your shoelace. Assuming you want to get to the other end as quickly as possible, is it better to tie your shoelace while you are on a moving walkway or while you are between walkways?
Some of my friends who work at the airport had more practical suggestions, such as "call a cart" or "wear loafers", but I am more interested in the mathematical question. The worksheet below shows a time-distance graph for each of the two scenarios. The green line shows what happens if you tie your shoelace while you are on the moving walkway, and the red line shows what happens if you tie your shoelace while you are between walkways.
Conclusion: It is quicker to tie your shoelace while you are on a moving walkway.
{airport shoelace.ggb}

Here is an intriguing real world math problem that was posed by Terry Tao.
You are walking from one end of an airport terminal to the other. The airport has several moving walkways, and you need to stop to tie your shoelace. Assuming you want to get to the other end as quickly as possible, is it better to tie your shoelace while you are on a moving walkway or while you are between walkways?
Some of my friends who work at the airport had more practical suggestions, such as "call a cart" or "wear loafers", but I am more interested in the mathematical question. The worksheet below shows a time-distance graph for each of the two scenarios. The green line shows what happens if you tie your shoelace while you are on the moving walkway, and the red line shows what happens if you tie your shoelace while you are between walkways.
{airport shoelace.ggb}