Constructions
ICSE Class 8 · Mathematics · Chapter 18
What this chapter covers
- Why a quadrilateral needs five measurements
- Tools and the basic constructions
- Four sides and one diagonal
- Three sides and two diagonals
- Three sides and two included angles
- Two adjacent sides and three angles
- Constructing special quadrilaterals
Key points
- A geometrical construction is an accurate drawing made with a ruler and compasses, with a protractor only for angles the compasses cannot make.
- A general quadrilateral is fixed by five independent measurements.
- Every construction splits the quadrilateral into two triangles and builds one after the other.
- Four sides and one diagonal: draw the diagonal, then build one triangle on each side of it.
- Three sides and two diagonals: draw the side shared by both triangles, then build both triangles on the same side of it.
- Three sides and two included angles: draw the middle side, make both angles on the same side, mark the other sides and join.
- Two adjacent sides and three angles: draw one side, make the angles in order, and let the last arm meet the first ray.
- Test every triangle (sum of two sides greater than the third) and every angle set (sum less than 360°) before you begin.
- Properties of parallelograms, rectangles, rhombuses and squares reduce the measurements needed to three, two, two and one.
- Always draw a rough sketch, leave construction arcs visible and write the steps of construction.
Questions answered
Why does a frame of four sticks change shape, while a frame with a fifth stick across it does not?
Four sides do not fix the angles, so the frame can lean. The fifth stick acts as a diagonal that splits the frame into two triangles, and a triangle with fixed sides cannot change shape.
How many independent measurements fix a general quadrilateral, and name the four standard cases.
Five. The four cases: four sides and one diagonal; three sides and two diagonals; three sides and two included angles; two adjacent sides and three angles.
In a construction with four sides and diagonal AC, why must B and D lie on opposite sides of AC?
You must draw a convex quadrilateral, and in a convex quadrilateral the two vertices not on a diagonal lie on opposite sides of it. If B and D were on the same side, two sides would cross or the figure would be concave with AC outside it, which is not the quadrilateral asked for.
Can a quadrilateral ABCD be drawn with AB = 2 cm, BC = 3 cm and diagonal AC = 6 cm?
No. 2 + 3 = 5, which is less than 6, so △ ABC cannot exist and the arcs never meet.
For AB, BC, CD with diagonals AC and BD given, which side do you draw first, and why?
BC, because it belongs to both △ ABC and △ BCD, whose three sides are all known.
Three angles of a quadrilateral are 75°, 105° and 120°. Find the fourth angle.
360° - (75° + 105° + 120°) = 360° - 300° = 60°.
Describe how to make an angle of 75° with compasses.
Construct 60° and 90° at the same vertex on the same ray, then bisect the angle between them.
A rectangle has a side of 5 cm and a diagonal of 4 cm. Can it be constructed? Give a reason.
No. In a rectangle, the diagonal forms the longest side of a right triangle, so it must be longer than 5 cm.