Construction II: Construction of Quadilaterals – SS1 Mathematics Lesson Note (Second Term)

To construct a quadrilateral with ruler and compasses, sketch it, draw one side, construct the given angle, and mark the remaining vertices with arcs of the given lengths. It also covers the equilateral triangle and loci of points moving at a given distance from a point or a line, or equidistant from two points or two lines.

Behavioural objectives

By the end of the lesson, students should be able to:

• construct a quadrilateral or parallelogram from given sides and an angle, and measure a diagonal;
• construct an equilateral triangle and bisect its angles;
• describe the locus of a point at a given distance from a fixed point or from a straight line;
• describe the locus of points equidistant from two given points or from two straight lines;
• construct loci in a triangle and measure the distance between their points of intersection.

How do you construct a quadrilateral?

Examples

1. Construct a quadrilateral ABCD in which AB is parallel to DC, |AB| = 4 cm, |BC| = 5 cm, |DC| = 7 cm and ∠ADC = 105°. Measure the diagonal BD.
2. Use your ruler and compasses to construct the parallelogram PQRS in which |QR| = 5 cm, |RS| = 11 cm and ∠QRS = 135°. Measure the length of the shorter diagonal of PQRS.

Solution to example 1

First sketch the quadrilateral to be constructed, then follow the steps below:

1. Draw line DNC = 7 cm with DN = 3 cm and NC = 4 cm.
2. Construct ∠CDM = 105°.
3. With N as the centre and a radius of 5 cm, draw an arc to cut DM at A.
4. With A as the centre and a radius of 4 cm, draw an arc.
5. With C as the centre and a radius of 5 cm, draw a second arc to cut the first arc at B.
6. Join A to B and C to B to complete the quadrilateral ABCD.

By measurement, |BD| = 4.5 cm.

Solution to example 2

First sketch the parallelogram PQRS, then follow the steps of the construction below:

1. Draw line QR = 5 cm.
2. Construct ∠R = 135°.
3. With R as the centre and a radius of 11 cm, draw an arc to cut the 135° line at S.
4. With S as the centre and a radius of 5 cm, draw an arc.
5. With Q as the centre and a radius of 11 cm, draw a second arc to cut the arc in step 4. This is point P.
6. Draw lines to join S to P and P to Q.
7. Draw a dotted line through diagonal RP and measure it.

By measurement, the length of the shorter diagonal PR is 8.3 cm.

Construction of an equilateral triangle

An equilateral triangle is a triangle in which all the sides are of equal length and each of its angles is 60°.

Examples

1. Construct an equilateral triangle XYZ such that |XY| = 5 cm.
2. (a) Construct an equilateral triangle ABC such that |AB| = 7 cm.
   (b) Construct the bisectors of ∠A, ∠B and ∠C.
   (c) What did you observe?

Construction of loci of moving points

Locus of points at a given distance from a fixed point

Draw a figure with O as a fixed point, and P1, P2 at a constant distance x cm from O. The locus of the points is a circle of radius x cm.

Locus of points at a given distance from a straight line

Draw a straight line AB which continues indefinitely in both directions. Add points P1, P2, P3 and P4 at a distance of x cm from AB. In two dimensions, the locus of the points consists of two straight lines parallel to AB, each at a distance x cm from AB.

Note that this locus consists of two separate lines.

Locus of points equidistant from two given points

Sketch a figure with X and Y as two fixed points. Points P1, P2 and P3 are such that |P1X| = |P1Y|, |P2X| = |P2Y| and |P3X| = |P3Y|. P1, P2 and P3 lie on the perpendicular bisector of XY. The locus of the points is the perpendicular bisector of XY.

Locus of points equidistant from two straight lines

Sketch a figure with straight lines AB and CD which intersect at O. P1 is equidistant from AB and CD. Similarly, P2 is equidistant from the two lines. P1 and P2 lie on the bisector of the acute angle between the two lines.

From the figure, it can be seen that P3 is equidistant from AB and CD. P3 lies on the bisector of the obtuse angle between the two lines.

Thus, the complete locus of points which are equidistant from two straight lines is the pair of bisectors of the angles between the lines (see the figure below).

Note that the two parts of the locus intersect at right angles.

Example

Using ruler and compasses only:

1. Construct triangle ABC such that |AB| = 6 cm, |AC| = 8.5 cm and ∠BAC = 120°.
2. Construct the locus l1 of points equidistant from A and B.
3. Construct the locus l2 of points equidistant from AB and AC.
4. Find the points of intersection, P1 and P2, of l1 and l2 and measure |P1P2|.

Solution

Note the construction of ∠BAC = 120°.

l1 is the perpendicular bisector of AB.

l2 is in two parts. AP1 is the bisector of ∠BAC. AP2 is perpendicular to AP1. Note that points on AP2 are equidistant from AB and CA produced.

By measurement, |P1P2| = 6.8 cm.

Evaluation

1. State the locus of a point which is always x cm from a fixed point O.
2. Describe the locus of points at a distance x cm from a straight line AB that continues indefinitely in both directions.
3. What is the locus of points equidistant from two given points X and Y?
4. Describe the complete locus of points equidistant from two intersecting straight lines.
5. Construct an equilateral triangle XYZ such that |XY| = 5 cm, and construct the bisectors of its angles.

Assignment

1. A circle, centre O, radius 5 cm, is drawn on a sheet of paper. A point P moves on the paper so that it is always 2 cm from the circle. The locus of P is:
   A. a circle, centre O, radius 3 cm
   B. two circles, centre O, radii 3 cm and 7 cm
   C. a circle, centre O, radius 6 cm
   D. two circles, centre O, radii 4 cm and 6 cm
   E. a circle, centre O, radius 3.5 cm
2. XYZ is a straight line such that |XY| = |YZ| = 3 cm. A point P moves in the plane of XYZ so that |PY| < |XY|. Which of the following describes the locus of P?
   A. a line through X perpendicular to XZ
   B. a line through Y perpendicular to XZ
   C. a line through Z perpendicular to XZ
   D. a circular disc, centre X, radius 3 cm
   E. a circular disc, centre Y, radius 3 cm
3. Describe the locus of a point which moves so that it is always 5 cm from a fixed point O in a plane.
   A. a rectangle which measures 10 cm by 5 cm
   B. a square of side length 5 cm
   C. a parallelogram whose diagonals are 10 cm and 5 cm
   D. a circle of radius 5 cm, centre O
   E. a circle of radius 10 cm, centre O
4. Describe the locus of a point which moves along a level floor so that it is 2 m from a wall of a room.
   A. One line, parallel to and 2 m from the wall
   B. Two lines, one on each side, parallel to and 2 m from the wall
   C. A circle of radius 2 m
   D. A semi-circle of radius ½ m
   E. Two perpendicular lines, each of length 2 m
5. Describe the locus of a point which moves so that it is 3 cm from a fixed line AB in a plane.
   A. 2 lines parallel to AB and 6 cm apart, joined by semi-circular ends
   B. 2 lines parallel to AB and 8 cm apart, joined by semi-circular ends
   C. 2 lines perpendicular to AB
   D. A circle of radius 6 cm
   E. A circle of radius 3 cm
6. Construct a trapezium ABCD in which AB is parallel to DC, |AB| = 4 cm, |BC| = 8 cm, |CD| = 11 cm and |DA| = 6 cm. (Hint: in a rough figure, divide the trapezium into parallelogram ABXD and triangle BCX. First construct triangle BCX.)
7. (a) Using ruler and compasses only, construct triangle ABC such that |AC| = 8.5 cm and ∠ACB = 135°.
   (b) Using any geometrical instruments, find a point P within triangle ABC which is at a distance 2.8 cm from AC and 6 cm from B. Measure the length of AP.