 Construction Instruments
 Construction of Perpendicular Lines
 Construction of Angles using a Ruler and a Pair of Compass Only
 Construction of Parallel Lines
 Proportional Division of lines
 Construction of Regular Polygons
 Construction of Irregular Polygons
 Past KCSE Questions on the Topic
Construction Instruments
The following minimum set of instruments is required in order to construct good quality drawings:
 Two set squares.
 A protractor.
 A 15 cm or 150 mm ruler
 Compass
 Protractor
 Divider
 An eraser/rubber
 Two pencils  a 2H and an HB, together with some sharpening device – Razor blade or shaper.
Construction of Perpendicular Lines
To obtain the perpendicular bisector PQ
The figure below shows PQ as a perpendicular bisector of a given line AB.
 With A and B as centre, and using the same radius,draw arcs on either side of AB to intersect at P and Q.
 Join P to Q.
To construct a perpendicular line from a point
The figure below shows PE, a perpendicular from a point P to a given line AB.
 To drop a perpendicular line from point P to AB.
 Set the compass point at P and strike an arc intersecting AB at C and D.
 With C and D as centres and any radius larger than onehalf of CD,
 Strike arcs intersecting at E.
 A line from P through E is perpendicular to AB.
To construct a perpendicular line from a point
 Using P as centre and any convenient radius,draw arcs to intersect the lines at A and B.
 Using A as centre and a radius whose measure is greater than AP,draw an arc above the line.
 Using B as the centre and the same radius,draw an arc to interact the one in (ii) at point Q.
 Using a ruler ,draw PQ.
Construction of perpendicular lines using a set square.
 Two edges of a set square are perpendicular.
 They can be used to draw perpendicular lines.
 When one of the edges is put along a line, a line drawn along the other one is perpendicular to the given line.
To construct a perpendicular from a point p to a line
 Place a ruler along the line.
 Place one of the edges of a set square which form a right angle along the ruler.
 Slide the set square along the ruler until the other edge reaches p.
 Hold the set square firmly and draw the line through P to meet the line perpendicularly.
Construction of Angles Using a Ruler and a Pair of Compass Only
The basic angle from which all the others can be derived from is the 60^{0}, 45^{0 }and 90^{0}
Construction of an Angle of 60^{0}
 Let A be the apex of the angle
 With centre A draw an arc BC using a suitable radius.
 With b as the centre draw another arc to intersect arc BC at D.
 Draw a line AE through D. The angle EAC is 600
Construction of an Angle of 90^{0}
 Let A be the apex of the angle.
 With centre A draw an arc BC of large radius.
 Draw an arc on BC using a suitable radius and mark it D.
 Using the same radius and point D as the centre draw an arc E.
 BD and DE are of the same radius.
 With centre D draw any arc F.
 With centre E draw an arc equal in radius to DF.
 Join AF with a straight line. Angle BAF is 900
Construction of an Angle of 45^{0}
 Draw AB and AC at right angles 45^{0 }to each other.With centre A and with large radius ,draw an arc to cut AB at D and and AC at E.
 With centres E and D draw arcs of equal radius to intersect at Draw a straight line from A through Angle BAF is 45^{0}
Construction of Angles of Multiple of 7½^{o}, 30^{0}, 15^{0}
The bisection of 60^{0 }angle produces 30^{0 }and the successive bisection of this angle produces 15^{0 }which is bisected to produce 7½^{o }as shown below
 Draw AB and AC at 600 to each other as shown above.
 With centre A, and a large radius, draw an arc to cut AB at E and AC at D.
 With centres E and D draw arcs of equal radius to intersect at F.
 Draw a line from A through F.
 Angle CAF is 30^{0 }half 60^{0}.
To Construct 15^{0}, 7½^{o}
 Draw AC and AF at 30^{0 }to each other as described above.
 With centres G and D draw arcs of equal radius to intersect at H
 Draw a line from A through H
 Angle CAH is 15^{0}.
 With centres J and D a further bisection can be made to give 7½^{0}
Construction of Parallel Lines
 To construct a line through a given point and parallel to a given line, we may use a ruler and a pair of compass only, or a ruler and a set square.
Using a Ruler and A Pair of Compass Only
Parallelogram method
The line EP parallel to AC is constructed as follows:
 With X as the centre and radius PQ, draw an arc.
 With Q as the centre and radius PX, draw another arc to cut the first arc at R.
 Join X to R.
Proportional Division of Lines
Lines can be proportionately divided into a given number of equal parts by use of parallel lines.
To divide line AB
 Divide line AB into ten equal parts.
 Through b, draw a line CB of any convenient length at a suitable angle with AB.
 Using a pair of compasses, mark off,along BC ,ten equal intervals as shown above.
 Join C to A.By using a set square and a ruler, draw lines parallel to CA.
 The line is therefore divided ten equal parts or intervals.
Construction of Regular Polygons
A polygon is regular if all its sides and angles are equal ,otherwise it is irregular.
Note:
 For a polygon of n sides,the sum of interior angles is ( 2n − 4 ) right angles.The size of each interior angle of the regular polygon is therefore equal to (^{(}^{2n−4)}/n) × 90^{o} .
 The sum of exterior angles of any polygon is 3600.Each exterior angle of a regular polygon is therefore equal to ^{360}/_{n}.
Construction of a Regular Triangle
 Draw AB, BC and CD equal in length to the sides of the required triangle
 With centre B and aradius AB draw the arc AF
 With centre C and radius CD draw the arc DG
 Where the arcs intersect at E is an apex of the triangle
 Join BE and CE with straight lines to form the triangle BCE
Construction of a Regular Quadrilateral
 Mark off one side of the square AB on the base line
 With centre A and radius AB draw the arc BC
 With centre B and radius AB draw the arc CD
 With centre E and radius AB step off F and G on arcs BC and AD respectively
 With centres E and F draw arcs of equal radius to intersect at H
 With centres E and G draw arcs of equal radius to intersect at J
 Erect perpendicular AH and BJ
 The arcs BC and AD cut the perpendiculars AH and BJ at K and L respectively
 To complete the square join K and L
Construction of a Regular Pentagon.
To construct a regular pentagon ABCD of sides 4 cm.
Each of the interior angles = (^{(10−4)}/_{5}) × 90 right angles = 108^{0}
 Draw a line AB = 4 cm long.
 Draw angle ABC = 108^{0} and BC =4 cm
 Use the same method to locate points D and E
Note;
 Use the same procedure to construct other points.
Construction of Irregular Polygons
Construction of Triangles
To construct a triangle given the length of its sides
 Draw a line and mark a point A on it.
 On the line mark off with a pair of compass a point B, 3 cm from A.
 With B as the centre and radius 5 cm, draw an arc
 With A as the centre and radius 7 cm,draw another arc to intersect the arc in (iii) at C.Join A to C and B to C.
To construct a triangle, given the size of two angles and length of one side.
Construct a triangle ABC in which ∠ BAC = 60^{0}, ∠ ABC = 50^{0 }and BC = 4 cm.The sketch is shown below.
 Draw a line and mark a point B on it.
 Mark off a point C on the line, 4 cm from B.
 Using a protractor, measure an angle of = 50^{0 }and = 70^{0} at B and C respectively.
To construct a triangle given two sides and one angle.
Given the lengths of two sides and the size of the included angle. Construct a triangle ABC, in which AB = 4 cm,BC =5 cm and ∠ ABC =60^{0}.Draw a sketch as shown below.
 Draw a line BC = 5 cm along
 Measure an angle of = 60^{0 }at B and mark off a point A, 4 cm from B.
 Join A to C.
To Construct a Trapezium.
The construction of a trapezium ABCD with AB = 8 cm ,BC = 5 cm, CD = 4cm and angle ABC = 60^{0 }and AB = 8 cm
 Draw a line AB = 8 cm.
 Construct an Angle of = 60^{0 }at B.
 Using B as the centre and radius of 5 cm, mark an arc to insect the line in (ii) at C.
 Through C , draw a line parallel to AB
 Using C as the centre and radius of 4 cm,Mark an arc to intersect the line in (iv) at D.
 Join D to A to form the trapezium.
Past KCSE Questions on the Topic
 Using a ruler and a pair of compasses only,
 Construct a triangle ABC in which AB = 9cm, AC = 6cm and angle BAC = 37½^{0}
 Drop a perpendicular from C to meet AB at D. Measure CD and hence find the area of the triangle ABC
 Point E divides BC in the ratio 2:3. Using a ruler and Set Square only, determine point E. Measure AE.

On the diagram, construct a circle to touch line AB at X and passes through the point C. (3 mks)  Using ruler and pair of compasses only for constructions in this question.
 Construct triangle ABC such that AB=AC=5.4cm and angle ABC=30^{0}. Measure BC (4 mks)
 On the diagram above, a point P is always on the same side of BC as A. Draw the locus of P such that angle BAC is twice angle BPC (2 mks)
 Drop a perpendicular from A to meet BC at D. Measure AD (2 mks)
 Determine the locus Q on the same side of BC as A such that the area of triangle BQC = 9.4cm^{2} (2 mks)
 Without using a protractor or set square, construct a triangle ABC in which AB = 4cm, BC = 6cm and ∠ABC = 67½^{0}. Take AB as the base. (3mks)
Measure AC.  Draw a triangle A^{1}B^{1}C^{1} which is indirectly congruent to triangle ABC. (3mks)
 Without using a protractor or set square, construct a triangle ABC in which AB = 4cm, BC = 6cm and ∠ABC = 67½^{0}. Take AB as the base. (3mks)
 Construct triangle ABC in which AB = 4.4 cm, BC = 6.4 cm and AC = 7.4 cm. Construct an escribed circle opposite angle ACB
 Measure the radius of the circle (5 mks)
 Measure the acute angle subtended at the centre of the circle by AB (1 mk)
 A point P moves such that it is always outside the circle but within triangle AOB, where O is the centre of the escribed circle. Show by shading the region within which P lies. (3mks)
 Using a ruler and a pair of compasses only, construct a parallelogram PQRS in which PQ = 8cm, QR = 6cm and PQR = 150^{0} (3 mks)
 Drop a perpendicular from S to meet PQ at B. Measure SB and hence calculate the area of the parallelogram. (5 mks)
 Mark a point A on BS produced such that the area of triangle APQ is equal to three quarters the area of the parallelogram (1 mk)
 Determine the height of the triangle (1mk)
 Using a ruler and a pair of compasses only, construct triangle ABC in which AB = 6cm, BC = 8cm and angle ABC = 45^{o}. Drop a perpendicular from A to BC at M. Measure AM and AC (4mks)
 Using a ruler and a pair of compasses only to construct a trapezium ABCD such that AB = 12 cm, ∠DAB = 60^{o}, ∠ABC = 75^{o}, and AD = 7 cm (5 mks)
 From the point D drop a perpendicular to the line AB to meet the line at E. measure DE hence calculate the area of the trapezium (5 mks)
 Using a pair of compasses and ruler only;
 Construct triangle ABC such that AB = 8cm, BC = 6cm and angle ABC = 30^{0}.(3 marks)
 Measure the length of AC
 Draw a circle that touches the vertices A, B and C.
 Measure the radius of the circle
 Hence or otherwise, calculate the area of the circle outside the triangle.
 Using a ruler and a pair of compasses only, construct the locus of a point P such that angle APB = 60^{0} on the line AB = 5cm. (4 mks)
 Using a set square, ruler and pair of compases divide the given line into 5 equal portions. (3mks)
 Using a ruler and a pair of compasses only, draw a parallelogram ABCD, such that angle DAB = 75^{0}. Length AB = 6.0cm and BC = 4.0cm from point D, drop a perpendicular to meet line AB at N
 Measure length DN
 Find the area of the parallelogram (10 mks)
 Using a ruler and a pair of compasses only, draw a parallelogram ABCD in which AB = 6cm, BC = 4cm and angle BAD = 60^{o}. By construction, determine the perpendicular distance betweenthe lines AB and CD
 Without using a protractor, draw a triangle ABC where ∠CAB = 30^{o}, AC = 3.5cm and AB = 6cm. measure BC
 Using a ruler and a pair of compass only, construct a triangle ABC in which angle ABC = 37.5^{o}, BC =7cm and BA = 14cm
 Drop a perpendicular from A to BC produced and measure its height
 Use your height in (b) to find the area of the triangle ABC
 Use construction to find the radius of an inscribed circle of triangle ABC
 In this question use a pair of compasses and a ruler only
 Construct triangle PQR such that PQ = 6 cm, QR = 8 cm and ∠PQR = 135°
 Construct the height of triangle PQR in (a) above, taking QR as the base
 On the line AC shown below, point B lies above the line such that ∠BAC = 52.5^{o }and AB = 4.2cm. (Use a ruler and a pair of compasses for this question)
 Construct ∠BAC and mark point B
 Drop a perpendicular from B to meet the line AC at point F . Measure BF
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