Construction

This unit draws three of them exactly, from a few given measures, with a ruler, a compass and a sharp pencil: the scalene quadrilateral (15.1), the trapezium (15.2) and the rhombus (15.3). You proved the rhombus’s properties in Unit 14; here they become the steps of its construction.

Every construction below is made of a few moves you learned in earlier grades. Here they are in one place.

A rough sketch first. Before you draw anything exactly, draw the figure freehand, name its corners in order round it, and write each given measure where it belongs. The sketch tells you which side to draw first, which point comes from which, and which of two crossing points you want. Examiners look for it.

A point from two lengths. If C must be 6 cm from A and 4.3 cm from B, draw an arc of radius 6 cm with the compass point on A, and an arc of radius 4.3 cm with the point on B. They cross at C. (Two arcs cross in two places, one on each side of AB: the sketch says which one.)

Angles with a compass. Without a protractor you can make these angles at a point, all from the 60∘° of an equilateral triangle:

Angle	How
60∘°	one arc from the point, then the same radius cut once along it
120∘°	the same radius cut twice along the arc
90∘°	bisect the angle between 60∘° and 120∘°
30∘°, 45∘°	bisect 0∘° and 60∘°; bisect 0∘° and 90∘°
75∘°, 105∘°	bisect 60∘° and 90∘°; bisect 90∘° and 120∘°
135∘°, 150∘°	bisect 90∘° and 180∘°; bisect 120∘° and 180∘°
15∘°	bisect 0∘° and 30∘°

Any other angle, such as 100∘°, is drawn with a protractor.

The perpendicular bisector. With a radius more than half of AB, draw arcs from A and from B, above and below the line. The line through the two crossings cuts AB at its midpoint, at right angles. Every point on it is the same distance from A as from B.

A line parallel to another. Two ways, both from Grade 8. At a point C on a line that crosses AB, make the co-interior angle: if the angle at B is 60∘°, make the angle at C 180∘° − 60∘° = 120∘°, on the same side; or copy the alternate angle, equal to it, on the other side. Either makes the new line parallel to AB.

A scalene quadrilateral has no special property: no equal sides, no parallel sides, no right angles that we know of. So nothing about it comes for free, and we must be given enough measures to fix it. How many?

A quadrilateral has four sides, four angles and two diagonals. Four sides alone are not enough, as the plot showed: a frame of four sticks leans and folds. But draw one diagonal, and the quadrilateral becomes two triangles. A triangle with three fixed sides can’t change shape, which is why roofs and bridges are built of triangles. So the rule for a quadrilateral is: five independent measures, enough to build it one triangle at a time. (The four angles count as three, since they always add up to 360∘°: given three, the fourth is known.)

The five measures can be given in three ways. Step through all three, then work each one yourself below.

(a) When four sides and a diagonal are given

(b) When two adjacent sides and three angles are given

(c) When three sides and two angles are given

Back to the plot

Kopila’s plot has sides 48 m, 43 m, 36 m and 42 m. They also measured the diagonal from corner A to corner C: 60 m. With that fifth measure the plot can be drawn exactly. At a scale of 1 cm to 10 m the measures become 4.8 cm, 4.3 cm, 3.6 cm, 4.2 cm and the diagonal 6 cm: Example (a) above.

The diagonal also gives the area: it splits the plot into two triangles whose three sides you know, and Heron’s formula Grade 9: Unit 5 gives the area of each.

For the triangle ABC, half the perimeter is $s = \frac{48 + 43 + 60}{2} = 75.5$ m, and its area is

$$\sqrt{75.5 \times 27.5 \times 32.5 \times 15.5} \approx 1022.7 \text{ m}^2$$

For the triangle ACD, $s = \frac{60 + 36 + 42}{2} = 69$ m, and its area is

$$\sqrt{69 \times 9 \times 33 \times 27} \approx 743.8 \text{ m}^2$$

The plot is about 1,767 m², close to 3½ ropani (a ropani is about 508.7 m²). Without the diagonal, the widget gave the same four sides any area from about 1,100 m² to 1,770 m².

Each answer shows the finished construction (blue lines are compass arcs, the quadrilateral shaded) and its steps, after a rough sketch you draw yourself. The check is a length or angle you can measure on your own drawing: within a millimeter or a degree is good work.

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

A trapezium is a quadrilateral with one pair of opposite sides parallel. In ABCD with $AB \parallel DC$, the parallel sides are drawn with matching arrows. Being parallel is one property the trapezium has for free, so it needs only four measures, not five: once AB and one corner C are placed, D must lie on the line through C parallel to AB.

So every trapezium construction has one extra move: draw the line through C parallel to AB, with a co-interior or an alternate angle, then find D on it.

(a) When three sides and an angle are given

(b) When all four sides are given and the parallel sides are marked

Four sides were not enough for a scalene quadrilateral; for a trapezium they are, because the parallel sides add a fact. The trick is a point E on AB with AE = DC.

(c) When three sides and one diagonal are given

(d) When two sides and two angles are given

Question 1

Question 2

Question 3

Question 4

A rhombus has so many properties that it needs only two measures. Every construction of it uses one or two of these facts:

• all four sides are equal;
• its diagonals bisect each other at right angles, so each diagonal lies on the perpendicular bisector of the other;
• each diagonal bisects the angles at its ends, and opposite angles are equal.

(a) When the lengths of two diagonals are given

(b) When a side and an angle are given

(c) When a side and a diagonal are given

(d) When a side and the angle between the side and a diagonal are given

Question 1: from the two diagonals

Question 2: from a side and an angle

Question 3: from a side and a diagonal

Question 4: from a side and the angle a diagonal makes with it

Grade 9’s final exam is set to CDC’s specification grid for Compulsory Mathematics, Grades 9–10, where Geometry is 13 marks in three questions. In the grid’s model paper the construction is part c of the rhombus question (question 12):

Part	Level	Marks	What it asks
a.	knowledge	1	a property of the figure
b.	understanding	1	an unknown from a property
c.	application	2	construct the figure from given measures

Part c wants the construction with its arcs showing; write its steps if there is time. Here is one of our own in that shape, on a trapezium.