Question Video: Determining the Center of a Circle Passing Through Two Points Mathematics • Third Year of Preparatory School

Video Transcript

In the figure, which point can be used as the center of a circle that passes through the two points 𝐴 and 𝐵?

Let’s recall that a circle can be mathematically defined as a set of points in a plane that are a constant distance from a point in the center. For example, if we had these two points 𝐶 and 𝐷, then with a point 𝑂 which is equidistant from both points 𝐶 and 𝐷, we could create a circle with center 𝑂 which passes through both points 𝐶 and 𝐷. Notice that the line segments 𝐶𝑂 and 𝐷𝑂 would both be radii of the circle.

In this problem, we need to consider which of the given points would be the center of a circle which passes through 𝐴 and 𝐵. To do this, we’ll need to consider which point is equidistant from 𝐴 and 𝐵. That means it’s the same distance away from both these points. If this was a question on paper, for example, we could get a ruler and measure the distance of points 𝑀, 𝑁, and 𝑃 from the points 𝐴 and 𝐵. The one which is equidistant from both points 𝐴 and 𝐵 would be the center of the circle. But let’s consider this a little more mathematically.

We can consider this line 𝐿 sub two. 𝐿 sub two forms a 90-degree angle with the line segment 𝐴𝐵. We can see from the marking on the line segment 𝐴𝐵 that the line 𝐿 sub two divides line segment 𝐴𝐵 into two congruent pieces. We can therefore say that line 𝐿 sub two is the perpendicular bisector of line segment 𝐴𝐵. The really useful thing about the perpendicular bisector is it also gives us all the points which are equidistant from two different points. And so, any point on the line 𝐿 sub two would be a point which is equidistant from 𝐴 and 𝐵. That means that we could pick any point on this line 𝐿 sub two and draw a circle which goes through 𝐴 and 𝐵. We also notice that of course the point 𝑁 lies on this line. We could even draw a part of the circle which has center 𝑁 and passes through 𝐴 and 𝐵.

And so, the answer is that point 𝑁 can be used as the center of the circle passing through 𝐴 and 𝐵. Any other point on this line would also work. We can observe that point 𝑀 would not work because the two line segments of 𝐴𝑀 and 𝐵𝑀 are not congruent. In the same way, point 𝑃 could not be the center of a circle because the line segments 𝐴𝑃 and 𝑃𝐵 are different, leaving us with the answer of point 𝑁.