Constructions and Loci
Use compasses and a straightedge to construct angle bisectors, perpendicular bisectors and perpendiculars. Describe loci and shade regions that meet distance conditions.
What a learner can do afterwards
- Construct angle bisectors, perpendicular bisectors and perpendiculars, leaving the construction arcs visible
- Describe loci set by distances from points, lines and line segments
- Shade the region that satisfies two conditions at once
1 · Read
An angle bisector splits an angle into two equal angles. Draw an arc centred at the vertex, crossing the arms at D and E. From D and E, draw equal-radius arcs meeting inside the angle. Join their meeting point to the vertex with a straightedge. The later setting may differ from the first arc's setting. Divide the whole angle by 2 to find a half; double a half to recover the whole. Bisecting a half again divides by 2 again.
A perpendicular bisector crosses a segment at its midpoint and at 90 degrees. Set the compass wider than half the segment. From both endpoints, use that same setting to draw arcs meeting on both sides of the segment. Join the two meetings. Every point on this line is equally far from the endpoints. Conversely, every point equally far from the endpoints lies on this line. Matching arc settings make the intersections equidistant.
Find a coordinate midpoint by averaging each coordinate separately. For A(1, 3) and B(7, 5), x = (1 + 7) / 2 = 4 and y = (3 + 5) / 2 = 4. Their midpoint is (4, 4). The perpendicular bisector passes through it. Other points can lie on the perpendicular bisector too.
Average the x-coordinates and y-coordinates separately.
To construct a perpendicular through P on or off a line, draw an arc centred at P crossing the line at two distinct points A and B. For P off the line, use a large enough radius. From A and B, draw arcs of equal radius greater than half AB, meeting on both sides. Join the intersections. This perpendicular passes through P because PA = PB. Keep all arcs to show the method. A ray inside a right angle splits its 90 degrees into two angles.
A locus is the set of all points satisfying a rule. Points exactly 2 cm from P form a circle boundary centred at P, with radius 2 cm. Points within 2 cm form the disc: the boundary and its interior. Points closer than 2 cm form only the interior. Here, within includes equality; closer than excludes it.
This outline shows a circle boundary, not a filled disc. The centre and radius are not marked.
Distance is the shortest distance. Within d cm of an infinite line gives a strip reaching d cm on each side, of width 2d. For a segment, use the perpendicular distance if its foot lies on the segment; otherwise use the nearest endpoint. Exactly d cm from a segment, for d > 0, forms a stadium boundary: two parallel segments joined by end semicircles. Within d cm includes its interior too.
For rules joined by and, keep only their overlap. Closer to A than B means A's side of the perpendicular bisector of AB, excluding the bisector. Within 4 cm of C gives C's disc, including its boundary. Q's side of m is an open half plane: exclude m, and Q must not lie on m. If m passes through a circle's centre, points strictly inside the circle on Q's side form a semicircular interior. Exclude its diameter and curved boundary.
Use matching compass settings for paired arcs. Check each distance rule and its boundary, then keep only points satisfying every condition.
Lesson, questions and flashcards written for this dataset.
Readings, questions and flashcards written for this dataset. Linked videos by Khan Academy.
More to read
Check the two compass stages
An angle-bisector construction uses two pairs of equal distances. The first arc puts D and E equally far from the vertex. The later arcs find a point equally far from D and E. Joining that point to the vertex gives the bisecting ray.
The two stages need not use the same compass radius. What matters in the later stage is using one radius for both D and E, chosen so the arcs meet inside the angle.
Keep the arcs visible when presenting the construction. They let another person check the centres, equal settings and meeting point. Measuring the resulting angles can check the drawing, but measuring alone does not show how it was constructed.
Use equal distances to find a perpendicular
The perpendicular bisector of AB is a whole line, not just the midpoint. Every point on it is equally far from A and B. This is why matching arcs from the endpoints work: their intersections supply two points on the required line.
Choose an endpoint radius greater than half AB, and use it at both endpoints. A smaller setting may leave no intersections; a setting of exactly half gives only one meeting point. Two distinct intersections determine the line.
To find a perpendicular through another point P, first use an arc centred at P to mark two points on the given line. P is equidistant from those points, so their perpendicular bisector passes through P.
Read the boundary rule
An exact-distance rule usually asks for a boundary. Exactly a positive distance from a point gives a circle. Including all smaller distances fills the disc. In this lesson, within includes the boundary, while closer than excludes it.
A line and a segment have different nearest points. A line continues indefinitely, so distance to it is perpendicular distance. A segment stops at its endpoints. Beyond an end, that endpoint can be the nearest point instead. This accounts for the semicircular ends of the segment's stadium-shaped distance boundary.
For two conditions joined by and, test both. A point can meet one rule and still be excluded by the other. In particular, a point on the perpendicular bisector of AB is equally far from A and B, so it is not closer to A.
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En anglaisTake it off screen
Where it sits
Learn first
Nothing; this is a starting point.
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.