Perpendicular Bisector - Interactive Activity

A perpendicular bisector cuts a segment into two equal halves at a right angle, and every point on it is the same distance from the segment's endpoints. These activities let you watch that distance property at work, use it to solve for missing lengths, and see where the three bisectors of a triangle meet.

Definitions and Main Ideas

A perpendicular bisector does two jobs at once: it is perpendicular to a segment and it passes through the segment's midpoint. Its key property is about distance — every point on the perpendicular bisector is the same distance from the two endpoints of the segment it bisects. In a triangle, the three perpendicular bisectors meet at the circumcenter, which is therefore the same distance from all three vertices.

Vocabulary

Perpendicular Bisector: A segment, ray, or line that is perpendicular to a segment at its midpoint.
Equidistant: The same distance from two or more points or objects.
Concurrent: Three or more lines or segments that intersect at a single point.
Circumcenter: The point where the perpendicular bisectors of the three sides of a triangle meet; it is equidistant from the three vertices of the triangle.

Main Ideas

Any point on the perpendicular bisector of a segment is equidistant from the segment's endpoints.
A perpendicular bisector does not have to come from a vertex of the triangle — it is built on a side, at that side's midpoint.
A triangle has up to three perpendicular bisectors, one for each side, and they are concurrent at the circumcenter.
The circumcenter is equidistant from the vertices of the triangle; it lies inside an acute triangle, outside an obtuse triangle, and on the hypotenuse of a right triangle.
Because the bisector passes through the midpoint, the two halves of the bisected segment are equal — a second equation alongside the equidistant property.

Equidistant Explorer

What to do: Slide the point along the perpendicular bisector and watch its two distances to the endpoints — they always match. Then slide a point off the bisector and watch the distances separate. Finally, switch to the triangle to see the three bisectors meet at the circumcenter.

Choose what to explore. The first two buttons use one segment and its perpendicular bisector; the third uses a whole triangle.

This slider moves point P up and down along the perpendicular bisector. Watch the two distances in the readout.

Circumcenter Explorer

What to do: Drag any vertex of the triangle and watch the three perpendicular bisectors follow. They always intersect at the circumcenter O, the center of the circle through all three vertices. Use the preset buttons for clean examples, then drag to change the kind of triangle and watch where O moves.

These buttons jump to a clean example of each kind of triangle. Drag any orange vertex handle afterward to explore from there, and watch where the circumcenter O moves.

Perpendicular Bisector Challenge

What to do: Answer each problem. Every figure shows a segment, its perpendicular bisector, and the segment's midpoint. A first miss gives a hint and another try; a second miss shows the answer and the worked solution.

The Next problem button cycles through all eight problems in order.

Build the Solution

What to do: Work each problem one step at a time. Step 1 chooses the statement that governs the problem; the later steps substitute and solve. Read each step's question, choose an answer, and use Continue to move on.

The Next problem button cycles through all five problems in order.

Your Turn

What to do: Answer each question. A first miss gives a hint and another try; a second miss shows the full explanation. Your score counts first-try answers only.

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Perpendicular Bisector — Mr. J Geometry. All activities on this page run entirely in your browser.