Perpendicular Bisectors and Circumcircle
A perpendicular bisector cuts a line segment in half at a right angle (90°). Every point on it is the same distance from both ends. In a triangle, the three bisectors meet at the circumcentre, the centre of its circumcircle.
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What Is a Perpendicular Bisector?
- A perpendicular bisector cuts a line segment in half.
- It meets the segment at a right angle () at the midpoint.
Drawing a Perpendicular Bisector
- With a compass, set the width greater than half the segment.
- Draw arcs above and below from both ends, keeping the same width.
- Draw a straight line through the two points where the arcs cross.
Key Property of a Perpendicular Bisector
- Every point on the perpendicular bisector is the same distance from both ends of the line segment.
- This helps you find points that are exactly in the middle between the two ends.
Perpendicular Bisectors and the Circumcentre
- The circumcentre is where the three perpendicular bisectors of a triangle meet.
- It is equal distance from all three vertices and is the centre of the circumcircle.
Practice Questions
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What is a perpendicular bisector?
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A perpendicular bisector does two jobs at once: it passes through the midpoint of the segment, cutting it into two equal parts, and it meets the segment at a right angle.
Drop either condition and it is no longer a perpendicular bisector. Every point on it is the same distance from the two endpoints of the segment.
Which tools do you use to construct the perpendicular bisector of a line segment?
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You construct a perpendicular bisector with a pair of compasses and a straight edge (ruler).
Keeping the compasses at one fixed width, more than half the segment, draw arcs from each endpoint; the arcs cross above and below the segment, and the ruler joins those two crossing points. The construction itself guarantees both the right angle and the two equal halves. No measuring with a protractor is needed.
How do you find the circumcentre of a triangle?
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The circumcentre is the point that is the same distance from all three vertices, so it must lie on the perpendicular bisector of every side.
Construct the perpendicular bisectors of the sides with compasses and a straight edge. Two of them are enough to fix the point where they cross; the third passes through the same point and acts as a check.
What is the circumcentre of a triangle?
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The circumcentre is the point that is the same distance from all three vertices of the triangle.
That distance is the radius of the circumcircle, the circle through all three corners, and the circumcentre is its centre. It is found where the perpendicular bisectors of the sides meet, and it can lie inside, on, or outside the triangle.
What does the circumcircle of a triangle pass through?
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The circumcircle is drawn with the circumcentre as its centre and a radius reaching out to a vertex.
Because the circumcentre is the same distance from all three vertices, that circle passes exactly through all three corners of the triangle. The circumcentre is the centre, so it sits inside the circle, not on it; the circle does not pass through the side midpoints, and it does not touch the sides. That is the inscribed circle, a different circle altogether.
You're building a triangular park and want to place a fountain equidistant from all three corners. Which geometric point should you find?
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"Equidistant from all three corners" means the same distance from the three vertices, and that point is the circumcentre, where the perpendicular bisectors of the sides meet.
The other centres do different jobs: the incentre is the same distance from the three sides (found with angle bisectors), the centroid is the balance point where the medians meet, and the orthocentre is where the altitudes from the vertices meet. Only the circumcentre gives equal distances to the corners.
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Perpendicular Bisectors and Circumcircle
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Perpendicular Bisectors and Circumcircle frequently asked questions
The questions students bump into most on this topic
Yes. The word perpendicular tells you it always meets the line segment at a right angle of 90 degrees. The word bisector tells you it also passes through the midpoint, cutting the segment into two equal halves. Both properties are always true together.
A line segment has exactly one perpendicular bisector. Only a single line can both pass through the midpoint and cross the segment at a right angle at the same time, so the perpendicular bisector is always unique for any given line segment you are working with.
Every point on a perpendicular bisector is the same distance from both ends of a side. Where two perpendicular bisectors cross, that point is equally distant from all three vertices, which is exactly the definition of the circumcentre of the triangle.
No. Constructing the perpendicular bisectors of just two sides is enough to find it. They cross at a single point, which is the circumcentre. If you draw carefully, the third bisector passes through the very same point, so two bisectors are always sufficient.
The circumradius is a length: the distance from the circumcentre to any vertex. The circumcircle is the circle itself, drawn with the circumcentre as its centre and the circumradius as its radius, passing through all three vertices of the triangle.
Imagine placing a barbecue grill so it sits the same distance from three friends. You find the circumcentre of the triangle formed by the three of you, place the grill there, then draw the circumcircle to show where any latecomers should sit.