Circle radius midpoints?

Question

radius of a circle has end points at O(-1,3) and R(2,2). Find two possible end points for the diameter that contains this radius. Describe any assumptions you make.

 

 

*I would also like to know how the answer was found, as the back of my book says the mindpoint of O is (-4,4) while R is (5,1).*

 

Thank you in advance :)

Answer 1

assumptions: since we know that a radius originates at the center of a circle and ends on the circle, then we know that one of the given points is the center and the other is on the circle. The diameter goes through the center, so the diameter in this problem will travel through whichever one is the center point. So there are 2 possible ways we can define this diameter. It either starts at O (on the circle) passes through R and ends at the other side (the right side) of the circle....OR it starts at R, passes through O, and ends on the circle to the left. A picture would make this much more clear.

 

Anyway...let's start with the case that the diameter starts at O and passes through R and continues (the same distance and trajectory) to the other side of the circle (definition of diameter). OK so the slope we see for the radius given points (-1,3) and (2,2) is the difference in y's over the difference in x's (definition of slope). The slope is -1/3 or simply, to get from one to the other we went down one and over 3. So we do that again to find the missing end point. Down one and over 3 would end us at (5,1).

 

The other case is that the diameter starts at R and continues through O to the other side of the circle (to the left). In this case we go up one and over 3 (to the left, negative). Only one value is negative when the slope is negative. You can use the negative value for the rise or the run but not both. If you used negative for both it would negate the negative and turn it into a positive slope (which it isn't so to keep the slope negative you only use one value negative). In this case to keep on the line we have to go up one and over (left) 3. That ends us at the point (-4,4)  

 

This is a problem that really needs a drawing and if I were tutoring you I would  show you how it would look on the xy plane. 

 

In your question you mention that your book says the "mindpoint" of O is (-4,4). I think you probably meant the "endpoint."