Question Video: Determining How Many Circles Pass Through Two Points | Nagwa Question Video: Determining How Many Circles Pass Through Two Points | Nagwa

Question Video: Determining How Many Circles Pass Through Two Points Mathematics • Third Year of Preparatory School

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How many circles can pass through two points?

03:38

Video Transcript

How many circles can pass through two points?

We can define a circle as a set of points in a plane that are a constant distance from a point in the center. Let’s imagine that these two points are here, and we can define them as 𝑃 sub one and 𝑃 sub two. These two points, 𝑃 sub one and 𝑃 sub two, will both lie on the same circle if they are the same distance from the center of the circle. For example, if we took this point 𝑀, could this be the center of a circle which passes through both the points 𝑃 sub one and 𝑃 sub two? And the answer would be no, because although the circle with center 𝑀 passes through 𝑃 sub one, it’s much too small to pass through 𝑃 sub two as well.

And so, to find a center of a circle which passes through 𝑃 sub one and 𝑃 sub two, we really need to consider this question: How do we find a point or a set of points, which are equidistant from two other points? Remember that this word “equidistant” simply means the same distance away. To do this, we’re going to construct the perpendicular bisector of the line segment between 𝑃 sub one and 𝑃 sub two. To do this accurately, we need this tool, which will be called a pair of compasses or a compass, depending on where you live. It can be helpful to draw in the line segment between the two points, and we start by putting the sharp pointed end of the compass into one of the points.

So, let’s start with 𝑃 sub one. We then stretch-open the compass so that the pencil tip will lie more than halfway along the length of the line segment. We then use the pencil in the compass to create an arc above the line segment and below it. We then repeat the process, this time putting the pointed end of the compass onto 𝑃 sub two. We now have a pair of arcs above the line segment and below it. We observed that in each pair of arcs there is a point of intersection. We join these two points of intersection with a straight line. This is the perpendicular bisector of the line segment between the two points. This line has divided the line segment between 𝑃 sub one and 𝑃 sub two into two congruent pieces and at 90 degrees.

However, in this problem, we’re not really interested in the line segment between the two points. But let’s consider this perpendicular bisector and what it actually represents. It will represent all the points that are equidistant from 𝑃 sub one and 𝑃 sub two. For example, here is a point on the line. Let’s call this point 𝐴. And because it’s equidistant from 𝑃 sub one and 𝑃 sub two, then we could create a circle of center 𝐴 and the circle with center 𝐴 will pass through both 𝑃 sub one and 𝑃 sub two. We could repeat this as many times as we wanted. For example, here’s a smaller circle and a larger circle. In fact, why even limit ourselves to circles which can fit on the screen? We know that this line will extend infinitely in both directions, and that means that we could draw an infinite number of circles.

And therefore for any two points, whether those points are really close together or thousands of kilometers apart, we know that we can draw an infinite number of circles that pass through those two points.

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