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SPI Mathematics · Lines, Angles and Polygons

Constructions

With a ruler and compass you can divide a segment in any ratio m to n by marking m plus n equal steps, and you can build 60 degrees from an equilateral triangle and halve any angle you have.

All Lines, Angles and Polygons notes

Why this matters

Five questions in the bank, four EASY and one HARD. Nobody asks you to draw: they ask how many points to mark on the ray, or which angle a set of arcs makes, so learn the steps well enough to count them.

Concept 1 of 2: Dividing a segment in a ratio

To cut AB in the ratio m to n, draw a slanting ray from A and mark m plus n equal steps on it. Join the last mark to B and draw a parallel line through mark m. Parallel lines cut the two lines in the same ratio, so the parallel meets AB at the point you want.

Definition

  • Draw a ray AX making an acute angle with AB.
  • Mark m+nm + n equal steps A1,A2,…,Am+nA_1, A_2, \ldots, A_{m+n} on AX. This is the least number of points needed.
  • Join Am+nA_{m+n} to B, and through AmA_m draw a line parallel to it. It meets AB at P, and AP:PB=m:nAP : PB = m : n.
  • A ratio m:nm : n splits the segment into m+nm + n equal parts, so the pieces are mm+n\frac{m}{m+n} and nm+n\frac{n}{m+n} of the length.

Dividing in the ratio m to n

points on the ray=m+nAP=mm+n×AB,PB=nm+n×AB\text{points on the ray} = m + n \qquad AP = \frac{m}{m+n} \times AB, \quad PB = \frac{n}{m+n} \times AB

Worked example

AB is to be divided in the ratio 3 : 4. How many equal points are marked on the ray AX, which point is joined to B, and through which point is the parallel drawn?
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Lines, Angles and Polygons · Constructions

To divide a line segment PQ in the ratio 5:7, first a ray PX is drawn so that QPX is an acute angle and then at equal distances points are marked on the ray PX such that the minimum number of these points is

Marking only the larger number of points

A ratio 5 : 9 needs 5+9=145 + 9 = 14 points, not 9. The ray must hold both parts of the ratio.

Marking one point too few

The steps are counted from A, so m+nm + n steps need m+nm + n marked points. 'm + n minus 1' is a printed wrong option.
Drill 2 more on dividing a segment in a ratio

Concept 2 of 2: Angles you can build with a ruler and compass

Two moves make every school construction: an equilateral triangle gives 60 degrees, and bisecting halves any angle you already have. Every angle in the table comes from those two moves.

Definition

  • 60 degrees: with the compass point on A draw an arc cutting the ray at D. With the same radius and the point on D, cut the arc at E. AD, AE and DE are equal, so triangle ADE is equilateral and the angle at A is 60∘60^\circ.
  • Bisecting an angle halves it, and can be repeated.
  • An angle with a half degree in it can still be built if it is half of an angle you can build.
AngleHow it is built
60°The equilateral arc construction
120°Two 60° steps along the same arc
90°Bisect the angle between the 60° and 120° marks
30°Bisect 60°
45°Bisect 90°
15°Bisect 30°
22½°Bisect 45°
135°90° plus half of the next 90°
67½°Bisect 135°
150°120° plus half of the last 60°
32½°Not by these steps
It is half of 65°, and 65° cannot be built from 60°, 90° and their halves.
If an angle comes from 60° and 90° by adding, subtracting and halving, you can build it.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Lines, Angles and Polygons · Constructions

On a ray AB with initial point A, taking A as centre and some radius, draw an arc of a circle, which intersects AB, say at a point D. Taking D as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point E. Draw the ray AC passing through E. Then, the measure of∠CAB\angle CAB is ____.

Thinking a half degree makes an angle impossible

2212∘22\frac{1}{2}^\circ is half of 45∘45^\circ and 6712∘67\frac{1}{2}^\circ is half of 135∘135^\circ, so both can be built.

Thinking every half angle can be built

Halving only works on an angle you can already build. 3212∘32\frac{1}{2}^\circ is half of 65∘65^\circ, and 65∘65^\circ does not come from 60∘60^\circ and 90∘90^\circ.
Drill 1 more on angles you can build with a ruler and compass

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

  • Dividing a segment in a ratio

    Dividing in the ratio m to n

    points on the ray=m+nAP=mm+n×AB,PB=nm+n×AB\text{points on the ray} = m + n \qquad AP = \frac{m}{m+n} \times AB, \quad PB = \frac{n}{m+n} \times AB

Reference tables (1)

Angles you can build with a ruler and compass11 rows
AngleHow it is built
60°The equilateral arc construction
120°Two 60° steps along the same arc
90°Bisect the angle between the 60° and 120° marks
30°Bisect 60°
45°Bisect 90°
15°Bisect 30°
22½°Bisect 45°
135°90° plus half of the next 90°
67½°Bisect 135°
150°120° plus half of the last 60°
32½°Not by these steps
It is half of 65°, and 65° cannot be built from 60°, 90° and their halves.
If an angle comes from 60° and 90° by adding, subtracting and halving, you can build it.

Watch out for (4)

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