A 90° angle at a point

The idea

To construct a 90° angle at a point O on a line, mark X and Y on the line equidistant from O; the perpendicular bisector of XY passes through O, so only one pair of arcs is needed.

A right angle at O

A point O is marked on a line. You want a 90° angle at O. You already know how to draw a perpendicular bisector.

How can the perpendicular bisector help?

• Mark X and Y on the line, the same distance from O, so that O is the midpoint of XY
• Measure 90° with a ruler
• Draw arcs only from O

Build the right angle

The first arc is drawn from O and cuts the line at X and Y. Then follow the steps. How many pairs of arcs will you need?

One point is already there

X and Y are the same distance from O, so O is the midpoint of XY. The perpendicular bisector of XY must pass through O. So O is one point of it already. We need only one more point, A, from one pair of arcs.

To construct a 90° angle at a point O on a line, mark X and Y on the line the same distance from O. The perpendicular bisector of XY passes through O, so only one pair of arcs is needed.

• Construction | Arcs needed
• Perpendicular bisector of XY | two pairs, giving A and B
• 90° at a point O | one pair: O and A fix the line

Notes

To construct a 90° angle at a point O on a line, mark X and Y on the line the same distance from O. The perpendicular bisector of XY passes through O, so one pair of arcs is enough.

Check yourself

Why is only one pair of arcs needed to draw the perpendicular at O?

O is the midpoint of XY, and OX = 3.5 cm. How many cm long is XY?

Answer: 7

O is the midpoint, so OY = OX = 3.5 cm and XY = 3.5 + 3.5 = 7 cm.

OX = OY = 2.5 cm. What is the smallest whole number of cm to which the compass can be opened so that the arcs from X and from Y meet?

Answer: 3

Half of XY is OX = 2.5 cm. The opening must be more than 2.5 cm, so the smallest whole number is 3 cm.

Which tools do you need to construct a 90° angle at O on a line?

• Because the arcs are drawn from O.. The arcs are drawn from X and Y. They meet at A. O is already on the bisector because it is the midpoint of XY.
• Because a right angle needs only one arc.. A line needs two points. Here O is one of them and A is the other.
• O is already a point of the perpendicular bisector of XY, so one more point A is enough. — correct. Yes! Two points fix a line: O and A.
• An unmarked ruler and a compass — correct. Yes! The ruler draws the line, and the compass makes the equal arcs.
• A protractor and a ruler. The construction does not measure the angle. Equal arcs make it exact.
• A set square and a protractor. Neither is needed. A ruler and a compass are enough.