Centres of a triangle · Ages 14–15
The perpendicular bisector's property
A point on the perpendicular bisector is equally far from both ends.
How to do it
Method
A point on the perpendicular bisector is equally far from both ends
AM = 4, PM = 3: PA = PB = 42 + 32 = 5
Another way
By congruent triangles
△PAM ≅ △PBM (SAS), PA = PB
In full
The perpendicular bisector theorem: any point on a segment's perpendicular bisector is the same distance from its two ends (⊥ bisector theorem). With M the mid-point of AB and PM ⊥ AB: PA = PB. The converse: a point equally far from the two ends lies on the perpendicular bisector (converse of ⊥ bisector theorem). One way: the theorem straight. PA = 7 cm gives PB = 7 cm. Another: congruent triangles. AM = BM, ∠PMA = ∠PMB = 90° and PM is common, so △PAM ≅ △PBM (SAS) and PA = PB. Or Pythagoras: AM = 4 cm and PM = 3 cm give PA = 42 + 32 = 5 cm = PB. △PAB is isosceles, its base angles equal.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In the picture, PM is the perpendicular bisector of AB. x =
Show the answer and working
Answer: 13 cm
P is on AB's perpendicular bisector, so it is equally far from A and B: x = PA = 13 (⊥ bisector theorem).
Example 2
P is on the perpendicular bisector of the segment AB, with PA = (2x + 1) cm and PB = (3x − 5) cm. x =
Show the answer and working
Answer: 6
PA = PB (⊥ bisector theorem): 2x + 1 = 3x − 5, so x = 6. Check: PA = PB = 13 cm.
Example 3
A point P is PA = 3 cm and PB = 5 cm from the ends of the segment AB. P lies on the perpendicular bisector of AB.
True or false?
Show the answer and working
Answer: False
Only a point equally far from both ends lies on the perpendicular bisector (converse of ⊥ bisector theorem).
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.P lies on the perpendicular bisector of the segment AB, and PA = 5 cm. PB =
Answer:
2.In the picture, PM is the perpendicular bisector of AB. So PA = PB ( ? ). Which reason goes in the gap?
- A. ⊥ bisector theorem
- B. prop. of isos. △
- C. ∠ bisector theorem
- D. converse of ⊥ bisector theorem
Choose the right answer
Answer:
3.PM is the perpendicular bisector of the segment AB, M on AB, with AM = 7 cm and PM = 24 cm. PA = cm, PB = cm
Fill in each blank
Answer:
4.P is on the perpendicular bisector of the segment AB, with PA = (2x + 1) cm and PB = (4x − 3) cm. x =
Answer:
5.A point P is PA = 3 cm and PB = 4 cm from the ends of the segment AB. P lies on the perpendicular bisector of AB.
True or false?
Answer:
All the answers
- 1. 5 cm
- 2. A. ⊥ bisector theorem
- 3. 25, 25
- 4. 2
- 5. False
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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