Use Equidistance from a Segment’s Endpoints
A segment’s perpendicular bisector crosses its midpoint at a right angle.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Point P lies on the perpendicular bisector of segment AB. PA=4 cm. Find PB. Candidate answers: (A) 0 cm (B) 2 cm (C) 4 cm (D) 8 cm.
- Independent check inputs: Point P lies on the perpendicular bisector of segment AB. PA=6 cm. Find PB. Candidate answers: (A) 0 cm (B) 12 cm (C) 3 cm (D) 6 cm.
Learn the idea
Congruence fixes a figure’s shape and size, while correspondence identifies matching vertices and sides. Perpendicular and angle bisectors encode equidistance or equal-angle properties that can be used and checked in a construction. A segment’s perpendicular bisector crosses its midpoint at a right angle. Every point on that line has equal distances to the segment’s two endpoints.
Here is the method on the supplied teaching case. Every point on a segment’s perpendicular bisector is equally distant from its two endpoints. Consequently PB=PA=4 cm. The theorem concerns endpoint distances, not distance to the midpoint.
Distance to either endpoint differs from distance to the midpoint. The theorem requires the point to be on the perpendicular bisector, not merely on a perpendicular line somewhere through the segment.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Point P lies on the perpendicular bisector of segment AB. PA=4 cm. Find PB. Candidate answers: (A) 0 cm (B) 2 cm (C) 4 cm (D) 8 cm. Solution: Every point on a segment’s perpendicular bisector is equally distant from its two endpoints. Consequently PB=PA=4 cm. The theorem concerns endpoint distances, not distance to the midpoint. The final answer is 4 cm. Distance to either endpoint differs from distance to the midpoint. The theorem requires the point to be on the perpendicular bisector, not merely on a perpendicular line somewhere through the segment.
Check your reasoning
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Helpful tips
- A segment’s perpendicular bisector crosses its midpoint at a right angle.
- Distance to either endpoint differs from distance to the midpoint.
See the answer and explanation
6 cm
Every point on a segment’s perpendicular bisector is equally distant from its two endpoints. Consequently PB=PA=6 cm. The theorem concerns endpoint distances, not distance to the midpoint.
Take it further
Changed-input task: Point P lies on the perpendicular bisector of segment AB. PA=18 cm. Find PB. Candidate answers: (A) 0 cm (B) 18 cm (C) 36 cm (D) 9 cm. Show the mathematical reason, not only the selected answer. Expected result: 18 cm. Every point on a segment’s perpendicular bisector is equally distant from its two endpoints. Consequently PB=PA=18 cm. The theorem concerns endpoint distances, not distance to the midpoint.
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