Constructing with a ruler and compasses

Mathematics · Geometry · ages 12-14

Name ______________________ Date ____________

You need a protractor to construct an angle bisector accurately.
Circle one: True False
Which construction uses a ruler and compasses to drop a straight line down to meet another line at a right angle from an outside point?
- Perpendicular from a point to a line
- Angle bisector
- Perpendicular bisector of a segment
When constructing a perpendicular bisector of segment AB, where do you place the compass point to draw your arcs?
- At A, then at B, using the same width both times
- Only at the midpoint of AB
- At A, then widen the compasses before moving to B
To construct a perpendicular bisector, you draw an arc above and below the line from point A, then an arc above and below the line from point B, all with the same compass width. How many arcs do you draw in total?
Answer: ______________
Omar puts the compass point on vertex Y and draws an arc that crosses both arms of angle XYZ. What should he do next to bisect the angle?
- Keep the same width, draw an arc from each crossing point, and mark where those two new arcs meet
- Measure the angle with a protractor and divide by two
- Draw a straight line straight from X to Z
Farid needs a perpendicular from point P down to line AB, using only a ruler and compasses. What should he draw first?
- An arc centered on P that crosses line AB in two places
- A straight line from P straight down by eye
- An arc centered on a point already on line AB
Segment AB is 8 cm long. Aisha constructs the perpendicular bisector correctly. How many centimeters is the crossing point from A?
Answer: ______________
Lucia bisects angle ABC. Her first arc from B crosses the two arms at X and Y. She then draws an arc from X, but changes the compass width before drawing an arc from Y. Is the line through where those two arcs cross the true bisector?
- No, the arcs from X and Y must use the same width, or they won't cross at the true bisector point
- Yes, because both arcs still crossed somewhere
- Yes, because only the first arc from B needs to be accurate

Answer key

For grown-ups. Fold this page away before handing over the rest.

Constructing with a ruler and compasses W1-mt_ZrsqGVG-Wt-s1

False · Compass arcs find the exact halfway line. No protractor is needed.
Perpendicular from a point to a line · This construction is called a perpendicular from a point to a line. It meets the line at exactly 90 degrees.
At A, then at B, using the same width both times · You draw arcs from both endpoints using the same compass width. Where the arcs cross marks the perpendicular bisector.
4 · Two arcs from A and two arcs from B make four arcs. The two crossing points give you the bisector line.
Keep the same width, draw an arc from each crossing point, and mark where those two new arcs meet · The two new arcs cross at a point exactly halfway between the arms. A line from Y through that point is the bisector.
An arc centered on P that crosses line AB in two places · The first arc, centered on P, marks two points on AB. Those two points set up the rest of the construction.
4 · A perpendicular bisector cuts the segment exactly in half: 8 divided by 2 is 4 cm.
No, the arcs from X and Y must use the same width, or they won't cross at the true bisector point · The arcs centered on X and Y must match in width. Different widths can still cross, but not at the exact halfway point.