Question Video: Determining the Center of a Circle Passing Through Two Points | Nagwa Question Video: Determining the Center of a Circle Passing Through Two Points | Nagwa

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

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In the figure, which point can be used as the center of a circle that passes through the two points π΄ and π΅?

02:48

### Video Transcript

In the figure, which point can be used as the center of a circle that passes through the two points π΄ and π΅?

Letβ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 πΆ and π·, then with a point π which is equidistant from both points πΆ and π·, we could create a circle with center π which passes through both points πΆ and π·. Notice that the line segments πΆπ and π·π 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 π΄ and π΅. To do this, weβll need to consider which point is equidistant from π΄ and π΅. That means itβ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 π, π, and π from the points π΄ and π΅. The one which is equidistant from both points π΄ and π΅ would be the center of the circle. But letβs consider this a little more mathematically.

We can consider this line πΏ sub two. πΏ sub two forms a 90-degree angle with the line segment π΄π΅. We can see from the marking on the line segment π΄π΅ that the line πΏ sub two divides line segment π΄π΅ into two congruent pieces. We can therefore say that line πΏ sub two is the perpendicular bisector of line segment π΄π΅. 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 πΏ sub two would be a point which is equidistant from π΄ and π΅. That means that we could pick any point on this line πΏ sub two and draw a circle which goes through π΄ and π΅. We also notice that of course the point π lies on this line. We could even draw a part of the circle which has center π and passes through π΄ and π΅.

And so, the answer is that point π can be used as the center of the circle passing through π΄ and π΅. Any other point on this line would also work. We can observe that point π would not work because the two line segments of π΄π and π΅π are not congruent. In the same way, point π could not be the center of a circle because the line segments π΄π and ππ΅ are different, leaving us with the answer of point π.

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