Geometrical construction
Please attempt the following on your own.
I WILL, in this week’s lesson, present a review of geometrical construction.
The following is extracted from the syllabus:
■ Draw and measure angles and line segments accurately using the appropriate instruments.
■ Construct lines, angles and polygons using the appropriate instruments.
Parallel and perpendicular lines. Bisecting line segments and angles. Constructing a line perpendicular to another line, L, from a point that is not on the line, L. Triangles, quadrilaterals, regular and irregular polygons. Angles include 300, 450, 600, 900, 1200, and their combinations.
You are invited to use the above to guide your review and study. Please note the following:
1. You should ensure that you are comfortable with the use of the protractor to measure angles, the pair of compasses to draw arcs representing specific distances, and the determination of distances using the ruler.
2. It is vital that you learn the methods of constructing the five angles listed above, using the pair of compasses, pencil and ruler only.
3. All methods are linked to the fact that the three internal angles of an equilateral triangle are each 600. Bisecting this angle will result in the construction of a 300 angle.
4. Being competent in bisecting line segments and angles, and also constructing parallel and perpendicular lines, is critical to completing the exam-type questions.
5. Repeated practice of each activity must be done to build competence.
Other points to note are as follows:
1. Construction arcs and lines must be shown to indicate the methods used.
2. The figure representing the answer should be bold.
3. The measurement of lines and angles are accepted within a specific degree of accuracy, and so every effort must be made to be as accurate as possible. The use of the divider will help.
4. The construction of parallel and perpendicular lines is usually required to construct a rectangle or parallelogram.
5. If the question restricts you to only use a ruler, a pencil and a pair of compasses, then a protractor must not be used.
In executing a construction without a protractor, please note the following skills which are required:
1. Draw a line of a given length: The pair of compasses and ruler are required.
2. Construct an angle of 60o: Construct an equilateral triangle, each angle is 60o.
3. Construct an angle of 30o: Construct a 60o angle and bisect it.
4. Construct an angle of 120o: Construct a 60o angle and it is the adjacent angle on a straight line.
5. Construct an angle of 90o: Construct a 60o angle adjacent to a 30o angle
6. Construct an angle of 45o: Bisect the 90o angle
Please review and practise these skills.
In attempting an example, you are encouraged to sketch the figure and then plan the order of the steps which are required.
EXAMPLE 1
Using a pair of compasses and ruler only, construct triangle PQR, such that PQ is 7.5cm, angle RPQ = 60o and angle PQR = 45o. Measure QR.
EXAMPLE 2
(i) Using a ruler, a pencil and a pair of compasses, construct: a) A triangle ABC in which AB = 8cm, AC = 6cm and angle A = 60o.
b) The line segment CM, which is perpendicular to AB and meets AB at M. (ii) Measure and state the size of angle BCM
SOLUTION
The following is a sketch of the triangle (not drawn to scale):
EXAMPLE 3
Using a ruler, a pencil and a pair of compasses:
(i) Construct accurately the square PQRS, with side 6cm.
(ii) Construct an extension of your drawing in (i), the trapezium RPQY so that angle PQY = 120o. (iii) Hence, measure and state the length of QY.
By making a sketch of the figure as seen above, the recommended order is: draw PQ = 6cm, angles P and Q = 900, and PR & QS = 6cm. Join RS and, finally, construct PQY = 1200 and then extend RS to meet QY at Y.
QY = 6.8cm
Please attempt the following on your own.
PRACTICE EXAMPLES
(i) Using a ruler, a pencil and a pair of compasses:
■ Construct the perpendicular bisector of a line segment PQ, of length 5cm.
■ Construct angle DEF = 450.
■ Construct triangle ABC with AB = 5cm, angle ABC = 600, and angle BAC = 900.
(ii) Measure and state:
■ The length of AC.
■ The size of angle ACB.
While the construction lines above are not clearly visible, be sure that yours are as you complete the examples.