Technical Drawing · Lesson 2

Lines, angles, triangles, polygons, circles, inscribed and escribed figures

WAECNECO

What the syllabus asks

WAEC: WAEC/NECO Technical Drawing, Plane Geometry items 3 (division of lines in equal parts and proportion), 5 (angles: types and constructions), 6 (triangles, quadrilaterals, regular and irregular polygons), 7 (circles: parts, types, properties, constructions) and 9 (inscribed, circumscribed and escribed figures). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS…

NECO: WAEC/NECO Technical Drawing, Plane Geometry items 3 (division of lines in equal parts and proportion), 5 (angles: types and constructions), 6 (triangles, quadrilaterals, regular and irregular polygons), 7 (circles: parts, types, properties, constructions) and 9 (inscribed, circumscribed and escribed figures). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS…

What you will learn

Bisect lines and draw perpendiculars and parallels by construction.
Divide a line into equal parts and in a given ratio, explaining why the method works.
Construct standard angles without a protractor and bisect and copy angles.
Construct triangles and quadrilaterals from given data.
Construct regular polygons by exact methods and know the approximate methods.
State circle properties and use them to find centres and draw circles through points.
Define and construct inscribed, circumscribed and escribed figures.

Key points

Bisect a line with equal arcs of radius more than half its length.
Perpendiculars: from a point on, off, or at the end of a line; 3-4-5 check.
Equal parts and ratios: inclined line, equal steps, parallels.
60° by equilateral arc; bisect for 30°, 15°; combine for 75°, 105°.
Triangle constructions: SSS, SAS, ASA, perimeter and ratio, perimeter and base angles.
Incentre (angle bisectors), circumcentre (side bisectors), centroid (medians), orthocentre (altitudes).
Quadrilateral properties, especially of diagonals.
Regular polygons: interior (n − 2)180/n, exterior 360/n.
Hexagon side = radius; exact pentagon method; some methods are approximate.
Circle properties: chord bisector through centre; tangent ⊥ radius; angle in semicircle 90°.
Inscribed inside, circumscribed around, escribed outside.
Keep construction lines visible; check by a second route.

Start here: Lines and the perpendicular bisector

Plane geometry is the construction of flat figures with compasses and a straight edge, using exact geometric reasoning rather than measuring with a protractor. WAEC Paper 2 and the plane-geometry part of Paper 1 depend heavily on it, and every later topic (tangency, loci, projection, development) uses these basic constructions. Draw all construction arcs lightly and leave them visible: they show the examiner your method.

Basic terms

A point has position but no size; it is shown by a small dot or the crossing of two short lines.
A line (straight line) is the shortest path between two points; a line segment AB has two ends.
Parallel lines are the same distance apart everywhere; perpendicular lines meet at 90°.
A locus is the path traced by a point moving under a rule (td4).

Construction 1: bisect a line AB

Set the compasses to a radius greater than half of AB.
With centre A, draw arcs above and below AB.
With centre B and the same radius, draw arcs cutting the first two at P and Q.
Join PQ. It cuts AB at M, the midpoint, and PQ is perpendicular to AB.

Why it works. P is the same distance from A and from B (both equal the radius), and so is Q. The set of points equidistant from A and B is the perpendicular bisector of AB; two points fix that line. Triangles APQ and BPQ are congruent (SSS), so PQ meets AB at right angles at its midpoint.

Construction 2: perpendicular from a point on a line

Given point C on a line: with centre C draw an arc cutting the line at D and E (CD = CE). Bisect DE as above; the bisector passes through C and is the required perpendicular.

Construction 3: perpendicular from a point off the line

Given point P above a line: with centre P draw an arc cutting the line at D and E. With centres D and E and equal radii, draw arcs crossing below the line at F. Join PF; it is perpendicular to the line.

Construction 4: perpendicular at the end of a line

Useful when there is no room to extend the line. At end B: choose any point O above the line, draw a circle with centre O through B, cutting the line again at D. Draw the diameter from D through O to meet the circle at E. BE is perpendicular to the line, because the angle in a semicircle is 90°. An alternative is the 3, 4, 5 method: mark 3 units along the line and 4 units up; if the diagonal measures 5, the angle is a right angle (3² + 4² = 5²). Builders setting out a foundation in Owerri use exactly this with a tape and pegs.

Continue this lesson in the free app →

Practice questions

1. To bisect a straight line you need

compasses and a straight edge
a protractor only
a 45° set square only
dividers only

Show answer

A: compasses and a straight edge
Arcs from both ends above and below give the bisector.

2. Each interior angle of a regular hexagon is

108°
135°
60°
120°

Show answer

D: 120°
(6 − 2) × 180 ÷ 6 = 120°.

3. Each interior angle of a regular pentagon is

72°
100°
108°
120°

Show answer

C: 108°
(5 − 2) × 180 ÷ 5 = 108°.

4. The side of a regular hexagon inscribed in a circle of radius 40 mm is

80 mm
40 mm
20 mm
69.3 mm

Show answer

B: 40 mm
The side of an inscribed hexagon equals the radius.

5. The side of a regular pentagon inscribed in a circle of radius 50 mm is about

58.8 mm
50 mm
62.8 mm
72 mm

Show answer

A: 58.8 mm
2 × 50 × sin 36° ≈ 58.8 mm.

Key words

Bisect: Divide into two equal parts.
Perpendicular: At 90° to a line.
Perpendicular bisector: Line at 90° through the midpoint of a segment.
Intercept theorem: Parallels cut two lines in the same ratio.
Complementary angles: Angles adding to 90°.
Supplementary angles: Angles adding to 180°.
Equilateral: Having all sides equal.
Isosceles: Having two equal sides.
Scalene: Having no equal sides.
Incentre: Meeting point of angle bisectors; centre of incircle.
Circumcentre: Meeting point of side bisectors; centre of circumcircle.
Centroid: Meeting point of medians.

Learn Lines, angles, triangles, polygons, circles, inscribed and escribed figures step by step, free.
The Cara's Tips app teaches every topic from the very basics, with quick checks, worked examples and exam practice. It works offline.
Open this lesson in the free app Download for Android