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It is given that points A = (−2, 1) B = (4, −1) form a segment, and it is asked at which point in y axis (when x = 0) when you "look at" or draw segments to points A and B there is a 90 degree angle. How do you solve this? The answer is 0,3 and 0,-3 but I cant understand how to find them.

I added a picture, it is known that x = 0, and is asked y coordinate, and the angle between red lines is 90. what is the coordinate of point c?

Picture of problem

Arthur
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Jeekim
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  • Use distance formula and Pythagoras theorem – 19aksh Dec 09 '20 at 15:13
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    https://en.wikipedia.org/wiki/Thales'_theorem – Arthur Dec 09 '20 at 15:14
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    So we are making the assumption here that the 90 degree angle must be at C? Because if the angle is allowed at A or B, there are more options. It is not explicitly clear to me that the angle must be at C – imranfat Dec 09 '20 at 15:20

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One way to do it, might not be the best way, using the fact that the product of the slope of perpendicular line is $-1$.

Let $A(-2,1)$, $B(4,-1)$ and $C(0,c)$ be three points. First, the slope of the line between $A$ and $C$ is $$m_1=\frac{c-1}{0-(-2)} = \frac{c-1}{2}$$ Second, the slope of the line between $B$ and $C$ is $$m_2=\frac{c-(-1)}{0-4} = \frac{c+1}{-4}$$ The product of the slopes must be $-1$ $$m_1\times m_2=\frac{c-1}{2}\times\frac{c+1}{-4} = \frac{c^2-1}{-8} = -1$$ This gives a quadratic equation $$c^2-1=8$$ which gives $$c=\pm3$$

Alain Remillard
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