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By-Hand Skills Sheet 2 — Constructing Triangles

Cambridge IGCSE Mathematics 0580 Extended · E4.2 · Name: ____________________   Date: __________
Constructions are marked on what the compasses, ruler and pencil leave on the paper — and the evidence is the arcs. This sheet is all-new practice in constructing triangles from their measurements. Print it and work on the printout; the model answers are hidden in print, so the sheet comes out blank to work on.

In the print dialog choose A4 and Scale: 100% — not “Fit to page”. Then work through it with a sharp pencil, a ruler, a protractor, a pair of compasses and an eraser, exactly as you would in the exam. Check your answers on screen afterwards: the model answers below are hidden when you print, so the sheet comes out blank to work on.

What you need before you start. A pair of compasses that holds its setting (test it — a loose pair is worse than none), a sharp HB pencil, a 30 cm ruler, a protractor and an eraser. Nothing on this sheet needs a calculator, and every length is in scale-bar units you can handle in your head.
About the scale bars. Some diagrams carry a printed scale bar marked in units. Set your compasses against that bar, not against a real centimetre ruler — if the printer has scaled the page even slightly, real centimetres will be wrong and the scale bar will still be right. In the exam the diagram is printed accurately and a real ruler is correct, so this is the one place this sheet differs from a real paper.

Section 1  ·  Constructing triangles (E4.2)

Compasses set against the scale bar for every length; protractor only where an angle is given. Every triangle ends with a measurement — that is how Cambridge checks a construction without watching you do it.
1.1 Using a ruler and compasses only, construct triangle ABC with AB = 7 units, AC = 5 units and BC = 6 units. AB is drawn for you. Set your compasses against the scale bar. Then measure the largest angle of your triangle with a protractor and write it down. [4]
All three sides are given, so this is a ruler-and-compasses construction: an arc of radius 5 from A, an arc of radius 6 from B. The largest angle always sits opposite the longest side.
A B 7 units 0 4 units
How the marks are awarded B1 arc of radius 5 units centred on A. B1 arc of radius 6 units centred on B, with C at the crossing — the arcs are what score: a perfect-looking triangle with no arcs looks measured and loses both marks. B1 sides AC and BC ruled in. B1 largest angle named at C, opposite the 7-unit side, measured about 78° (accept 76° to 81°).
A B C largest angle ≈ 78° arcs left in 0 4 units
  • Marking points
  • Set the compasses to 5 units against the scale bar, not a centimetre ruler, and arc from A. Reset to 6 units and arc from B. C is where the arcs cross.
  • Leave both arcs in. They are the evidence that C was constructed rather than guessed, and they carry the first two marks.
  • The largest angle faces the longest side, so it is at C. Cosine-rule check, no calculator needed: cos C = (25 + 36 − 49) ÷ 60 = 12⁄60 = 0.2, so C is a little under 80°.
  • A protractor reading of 76°–81° earns the mark. Far outside that range means the compass setting drifted between arcs.
1.2 Construct triangle PQR with PQ = 6 units, angle QPR = 40° and PR = 4.5 units. PQ is drawn for you. Use a protractor for the angle and compasses set against the scale bar for PR. Then measure QR against the scale bar and write it down. [4]
The given angle is at P, between the two given sides. Draw the 40° ray first, then mark 4.5 units along it with one compass arc.
P Q 6 units 0 4 units
How the marks are awarded B1 ray from P at 40° to PQ (protractor is allowed here because the angle is given). B1 compass arc of radius 4.5 units from P cutting the ray at R — that arc scores; marking 4.5 with a ruler against real centimetres does not, because the printout may be scaled. B1 QR ruled. B1 QR measured against the scale bar: about 3.9 units (accept 3.7 to 4.1).
P Q 40° R measure: about 3.9 units 0 4 units
  • Marking points
  • Angle first, length second: draw a long ray at 40° from P, then one arc of radius 4.5 units cuts R onto it. Leave the arc in.
  • The protractor is legal here because 40° is given. It is never legal when the question says compasses only.
  • To measure QR, set your compasses to the gap QR, then read that opening off against the scale bar.
  • QR should come out close to 3.9 units (accept 3.7 to 4.1). If yours is nearer 9.9, you read the wrong scale on the protractor and drew 140° instead of 40°.
1.3 Construct triangle XYZ with XY = 7 units, angle ZXY = 35° and angle ZYX = 65°. XY is drawn for you. Then measure XZ against the scale bar and write it down. [4]
Both given angles sit on the given side XY, one at each end. Draw both rays long — Z is wherever they cross, and short rays that stop early are the classic time-waster.
X Y 7 units 0 4 units
How the marks are awarded B1 ray from X at 35°. B1 ray from Y at 65°, both measured with a protractor (allowed — the angles are given), with Z at the crossing. B1 triangle completed with ruled lines. B1 XZ measured against the scale bar with compasses: about 6.4 units (accept 6.2 to 6.6). Compass arcs used to transfer the length onto the scale bar score — a real-centimetre ruler reading may not, if the print is scaled.
X Y 35° 65° Z about 6.4 units 0 4 units
  • Marking points
  • Both base angles open towards each other: 35° above XY at X, 65° above XY at Y.
  • Rule each ray well past where you expect Z. If they cross off the top of your space, the rays were drawn on the wrong sides.
  • The third angle is 180 − 35 − 65 = 80° at Z — a quick head-check before you measure anything.
  • XZ faces the 65° angle. Expect about 6.4 units; transfer the length with compasses to the scale bar to read it.
1.4 In triangle KLM, KL = 8 units, LM = 5 units and angle KLM = 55°. KL is drawn for you. Work out which vertex the 55° angle is at, construct the triangle, then measure KM against the scale bar. [4]
Read the angle’s name: the middle letter of angle KLM is the vertex where the 55° goes. Then ask which of the two given sides start there.
K L 8 units 0 4 units
How the marks are awarded B1 the 55° drawn at L, the middle letter of angle KLM, opening back over the base towards K (protractor allowed — the angle is given). B1 compass arc of radius 5 units from L, set against the scale bar, cutting the ray at M — the arc scores, so leave it in. B1 KM ruled to complete the triangle. B1 KM measured against the scale bar: about 6.6 units (accept 6.4 to 6.8).
K L 55° M about 6.6 units the 55° sits at L, between the two given sides 0 4 units
  • Marking points
  • The 55° is at L, the middle letter of angle KLM, and the two known sides KL and LM both start at L — so the angle sits between them.
  • The angle opens from ray LK, so the new ray leans back over the base — do not draw it opening away past L.
  • One arc of radius 5 units (set against the scale bar) from L fixes M on the ray. Leave it visible.
  • KM should measure about 6.6 units. If you got about 11.6, the angle was drawn opening the wrong way.

How examiners award the marks on this sheet

Read this before you start, and again before you hand the paper in.
  1. Never rub out a construction arc. The arcs are the method mark. A perfect line with no arcs looks exactly like a measured line, and measured lines score nothing when the question says ruler and compasses only.
  2. Set lengths against the scale bar, transfer them with compasses, and read measurements back the same way. On this printed sheet the bar is right even when real centimetres are not.
  3. Protractor rules: allowed when an angle is given (as in 1.2, 1.3 and 1.4), banned when the question says compasses only, and always fine for checking.
  4. Rule every straight line and draw every ray long. When two angles are given, the third vertex is wherever the two rays cross, and a ray that stops early never gets there.
  5. Let the mark allocation count your evidence. A 4-mark triangle wants four visible things. Count what is on the page against the marks before you move on.
Cambridge IGCSE Mathematics 0580 Extended · by-hand skills sheet 2 · 4 exercises · 16 marks · E4.2 constructing triangles · Print at 100% on A4 and complete in pencil.