Bisector and perpendicular bisector

The idea

Dividing a line, or any geometrical object, into two identical parts is bisection; a line that bisects a given line segment and is perpendicular to it is its perpendicular bisector.

The eye and its lines

In Grade 6 you drew an eye with two arcs. The dotted line XY supports it. The dotted line AB joins the two centres of the arcs. They cross at O.

What do you notice about where AB meets XY?

• AB meets XY in its middle, at a right angle
• AB meets XY near X, at a slant
• AB does not meet XY at all

Two things a line can do

A line crosses XY at P. Slide P along XY and turn the line. See when it bisects XY and when it is perpendicular to XY.

Bisect, and then perpendicular

To bisect is to divide into two identical parts. A line bisects XY when it passes through its midpoint. It can be tilted. Many lines bisect XY.

Bisection works for other objects too: a rectangle cut into two identical halves is bisected.

Dividing a line, or any geometrical object, into two identical parts is bisection. A line that bisects a line segment and is perpendicular to it is the perpendicular bisector of the segment.

• The line | It is
• through the middle, tilted | a bisector only
• at 90°, but not in the middle | perpendicular only
• through the middle, at 90° | the perpendicular bisector

Notes

Dividing a line, or any geometrical object, into two identical parts is bisection. A line that bisects a segment and is perpendicular to it is the perpendicular bisector.

Check yourself

A line passes through the midpoint of XY but meets it at 60°. Is it the perpendicular bisector of XY?

XY is 10 cm long. A line meets XY at 90°, 3 cm from X. How many cm from X must a perpendicular meet XY so that it bisects XY too?

Answer: 5

The midpoint is half way along: 10 ÷ 2 = 5 cm from X.

Many lines pass through the midpoint O of XY. How many of them are perpendicular to XY?

A square sheet is folded along a line so that one half lies exactly on the other half. Does the fold line bisect the square?

• Yes. Passing through the midpoint is enough.. That only makes it a bisector. It must also be perpendicular, at 90°.
• No. It bisects XY, but it is not perpendicular to it. — correct. Yes! It needs both: through the midpoint and at 90°.
• No. A bisector must not pass through the midpoint.. A bisector passes through the midpoint. This one fails only because the angle is 60°, not 90°.
• All of them. Only the line at 90° is perpendicular. The tilted ones are bisectors only.
• None of them. One of them is at 90°: the perpendicular bisector.
• Only one — correct. Yes! Of all the lines through O, only one makes a 90° angle with XY.
• Yes. It divides the square into two identical parts. — correct. Yes! Two parts that fit exactly on each other are identical, so the fold bisects the square.
• No. Only a line segment can be bisected.. The book says any geometrical object can be bisected, a square too.
• No. A fold is never a bisector.. A fold that puts one half exactly on the other divides the shape into two identical parts.