Equally far means on the bisector

The idea

A line segment has only one perpendicular bisector, and any point that is at the same distance from X and Y lies on the perpendicular bisector of XY.

More centres for the eye

One eye used the centres A and B on the line AB. Now choose two other points C and D, also with CX = CY = DX = DY, for a differently shaped eye.

Will C and D lie on the line AB?

• Yes, on AB
• No, they can be anywhere
• Only if C and D are at the same distance as A and B

Hunt for equal distances

Drag P around. The gauge shows how far P is from X and from Y. When PX = PY, P clicks onto a line. Pin such points, then use them as centres of eyes.

One line holds them all

Take C and D with CX = CY = DX = DY. The line joining them is the perpendicular bisector of XY (Lesson 2). But XY has only one perpendicular bisector: the line AB. So C and D are on AB.

A line segment has only one perpendicular bisector, and any point at the same distance from X and Y lies on it.

• The point P | Where it is
• PX = PY | on the perpendicular bisector
• PX is less than PY | on X’s side of it
• PX is more than PY | on Y’s side of it

Notes

A line segment has only one perpendicular bisector, and any point that is at the same distance from X and Y lies on it.

Check yourself

A point P is 3 cm from X and 3 cm from Y. Where is P?

A point Q is 2 cm from X and 5 cm from Y. Is Q on the perpendicular bisector of XY?

XY is 10 cm long. P is a point on its perpendicular bisector, and PX + PY = 26 cm. How many cm is PX?

Answer: 13

P is on the perpendicular bisector, so PX = PY. Together they are 26 cm, so PX = 26 ÷ 2 = 13 cm.

Meena says: “I drew two different lines that are both perpendicular bisectors of the same segment XY.” Can she be right?

• On the line XY itself. P could be there only if XY is 6 cm and P is its midpoint. In general it is on the perpendicular bisector.
• On the perpendicular bisector of XY — correct. Yes! PX = PY, so P lies on the perpendicular bisector.
• Nearer to X than to Y. It is exactly as near to X as to Y.
• Yes. Every point near XY is on the bisector.. Being near is not enough. The distances must be equal.
• Yes. It is on the bisector if its distances add up to 7 cm.. The sum does not matter. Only equal distances put a point on the bisector.
• No. It is nearer to X, so it is on X’s side of the bisector. — correct. Yes! Only points with equal distances lie on the perpendicular bisector.
• No. A line segment has only one perpendicular bisector. — correct. Yes! The two lines would have to be the same line.
• Yes, a segment has two: one above and one below.. It is one single line. The parts above and below XY belong to the same line.
• Yes, if the segment is long enough.. The length does not matter. There is exactly one.