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 is a line that cuts a segment into two equal parts at a right angle.
Which tool is used to construct the perpendicular bisector of a segment?
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The ruler helps us draw straight lines and measure lengths, while the compass allows us to create equal-length arcs from each endpoint of the segment. This helps us find the midpoint and draw a perpendicular line through it.
How do you find the circumcentre of a triangle?
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To find the circumcentre, you need to draw the perpendicular bisectors of at least two sides of the triangle. Their intersection point is the circumcentre.
What is the circumcentre of a triangle?
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The circumcentre of a triangle is the point that is equidistant from all three vertices of the triangle. It is the centre of the circle that passes through all three vertices (circumcircle).
What does the circumcircle of a triangle pass through?
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The circumcircle passes through all three vertices of the triangle. It is the circle whose centre is the circumcentre.
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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You need to find the circumcentre. The circumcentre is the point that is the same distance from all three vertices, making it the perfect location for the fountain.
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Interactive Activity
Perpendicular Bisectors and Circumcircle
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Students Also Ask
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.