4.5 Construction
- Syllabus
- 2017
- Topic
- 4.5
- Level
- Higher
Learning objectives
4.5A Measure and draw lines to the nearest millimetremeasure and draw lines to the nearest millimetre4.5B Constructing two-dimensional shapesconstruct triangles and other two-dimensional shapes using a combination of a ruler, a protractor and compasses4.5C Problems using scale drawingssolve problems using scale drawings4.5D Perpendicular and angle bisectorsuse straight edge and compasses to: (i) construct the perpendicular bisector of a line segment (ii) construct the bisector of an angle
Measure and draw lines to the nearest millimetre
Align the ruler's zero mark with one endpoint, keep its edge along the segment and read the other endpoint. Record to the nearest millimetre, including the unit.
| Task | Reliable check |
|---|---|
| measure | start at zero, not at the ruler's physical edge |
| draw | mark both endpoint positions before joining |
| convert | 10extmm=1extcm |
| report | nearest millimetre means the nearest 0.1extcm |
A reading of 5.37extcm is 53.7extmm, which rounds to 54extmm or 5.4extcm to the nearest millimetre.
If the starting endpoint is at the 2 cm mark rather than zero, subtract the two ruler readings; do not report the final mark itself as the length.
Construct triangles and other 2D shapes
A construction locates each unknown vertex as the intersection of exact length or angle conditions. Leave compass arcs and guide lines visible as evidence of the method.
| Given condition | Tool and action |
|---|---|
| fixed length from a point | compass arc with that radius |
| fixed angle at a vertex | protractor ray from the baseline |
| straight side | ruler through located vertices |
| two side lengths | intersect two arcs, one from each known endpoint |
For sides AB=8 cm, AC=6 cm and BC=9 cm: draw AB; draw an arc radius 6 cm centred at A and an arc radius 9 cm centred at B; their intersection is C; join AC and BC.
A neat sketch without arcs or angle guides does not demonstrate an exact construction. Keep the compass width fixed for each required radius.
Solve problems using scale drawings
A scale links a drawing length to a real length. Use the same scale factor in every direction, and attach the correct real-world unit after converting.
| Step | Action |
|---|---|
| 1 | write the scale as drawing : real |
| 2 | convert both lengths to compatible units |
| 3 | multiply or divide by the scale factor |
| 4 | for a position, combine the required distance with its direction or bearing |
At scale 1extcm:5extkm, a drawing distance of 3.6 cm represents 18 km. A real distance of 12 km is drawn as 12/5=2.4 cm.
A linear scale factor applies to lengths. Do not square it unless the question asks about area, and do not measure from the wrong starting point when plotting a location.
Construct perpendicular and angle bisectors
To bisect segment AB, use one compass radius greater than half of AB. Draw arcs above and below from both A and B; join the two arc intersections. This line is perpendicular to AB and passes through its midpoint.
To bisect an angle, draw one arc centred at the vertex to cut both arms. From those two cut points, draw equal-radius arcs that intersect; join the vertex to that intersection.
| Construction | Arcs that must remain visible |
|---|---|
| perpendicular bisector | equal-radius pairs centred at both endpoints |
| angle bisector | vertex arc plus equal arcs from its two arm intersections |
Points on a perpendicular bisector are equidistant from the segment endpoints. Points on an angle bisector are equidistant from the two angle arms.
A measured midpoint or protractor line is not a straight-edge-and-compasses construction. Equal compass radii create the required symmetry; do not alter the radius within a paired set of arcs.