Lesson Notes By Weeks and Term - Senior Secondary School 1

SUBJECT: MATHEMATICS

CLASS:� SS 1

DATE:

TERM: 3rd TERM

REFERENCE BOOKS

• New General Mathematics SSS 1 by M.F. Macrae et al�
• Essential Mathematics SS 1

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WEEK THREE

TOPIC: GEOMETRICAL CONSTRUCTION

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• Revision of Construction of Triangle�

• Drawing and bisection of Line Segment�

• Construction and Bisection of Angles 90o, 45o, 135o, 221/2o, 571/2o

• Construction and Bisection of Angles: 30o, 60o, 90o, 120o, 150o

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• Revision of Construction of Triangles

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Example

1. Construct � � � ABC in which !AB ! = 7cm, !AC! = 9.5cm and ABC = 120o.Measure !BC!
2. Construct� � � � � PQR in which !PQ != 5.5cm !QR! = 8..5cm and PQR = 75o . Construct M the midpoint of PR Measure /QM/

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Solutions

1. First make a sketch of the triangle to be constructed .

Draw a line AB = 7cm

Then construct angle 120o at B with radius 9.5cm and centre� A, draw an arc to cut the 120 o at�

1. Draw line AC

From the diagram /BC/ = 3.6cm

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2)� First make a sketch of the triangle to be constructed.

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5. Draw line PQ = 5.5cm
6. Construct angle 75o at Q
7. With centre Q and radius 8.5cm , draw an arc to cut the angle 75o at R.
8. Draw line QR.
9. Bisect line PR

From the diagram, /QM/ = 5.5cm

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EVALUATION

7. Construct � � � XYX in which /YZ/ = 7.5cm� XYZ = 60o and� XZY = 45o

b measure !XY !and !XZ!�

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• Drawing and Bisection of Line Segments

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To bisect a given line segment means to divide the given line segment into two parts of equal length.� The steps to take to bisect a given line segment are as follows:

1. Draw the� given line segment AB ( let AB = 8.6cm)
2. With centre A and radius of about � of length of AB, draw an arc above and below the line AB�
3. With centre B and the same radius used in step 2 above, draw arcs to cut the previous arcs in step 2.
4. Draw a line through the 2 points of inter- section of the pair or arcs obtained from steps 2 and 3 . The line drawn is the perpendicular bisector of line AB.

Thus AE� = EB = 4.3cm

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EVALUATION�

1. Draw a line CD = 11cm

Bisect� the line CD

2. Construct the mid point M of the line drawn below

Where length CD = 10.6cm

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1. Construction and Bisection of angles : 90o, 45o, 135o, 22 � , 67 � o.����������������������������������������������������

To construct angle 90o, take the following steps:���

1. Draw a line BC and mark a point A at which the angle 90o is to be constructed .
2. With centre B and any suitable radius draw� an arc above line BC.
3. With centre C and the same radius used in step 2, draw an arc to cut the previous Arc at D.
4. Draw a line through points A and D. thus < DAB = o

Since 45o = � of 90o, angle 90o is bisected to obtain angle 45o. This is shown in the figure below:

Thus < IGF = 45o

Also < HGI = 45o

Similarly 22 � o = � of 45o, By bisecting angle 45o, we can obtain angle� 22 � o as shown in the figure below:

Thus o.� Also o

Also 135o = 90o + 45o. Thus by constructing angle 90o at a point on a line and bisecting the 90o on the other side, we can obtain angle 135o. This is shown in the figure below:

Thus o.

As explained above bisection of angle 135o will give angle 67 � o

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Bisection of a given Angle.

The step to take for bisecting� a given angle are as follows.

1. Draw the given angle ABC i.e
2. With centre B and any suitable radius, draw an arc to cut AB at D and BC at E.
3. With centre D and any suitable radius, draw an arc
4. With centre E and the same radius as the one used in step (3) above, draw another arc to cut the previous arc at F.
5. Draw the line BF. Line BF is the bisector of ABC. This is shown in the figure below

Thus < FBC = , ABF = �

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EVALUATION�

1. Construct angle 135o

2a.� Construct angles 22 � o

1. � Construct angle 67 ��

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• Construction and bisection of angle: 60o, 30o, 75o, 105o, 120, 150o.

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To construct angle 60o, the following steps must be taken:

1. Draw a line AB and mark the point A at which the angle 60o is� to be constructed
2. With centre A and any convenient radius, draw an arc to cut line AB at C.
3. With centre C and the same radius used to draw the arc in step 2 above, draw another arc to cut the previous arc at D.
4. Draw line AD and extend it to E
5. Then ,EAB = 60o.

To construct angle 30obisect� angle 60o to give angle 30o, this� is shown in the figure below:

Thus, o

To construct angle 75o . Since 75o = 60 + � of 30o, then first construct angle 90o at a point on a straight line. Next construct angle 60o at the same point where angle 90o has been constructed.� Then the angle 30o difference between the angle 90o and 60o is bisected to give 15o on either side .thus 60 o + 15 = 75.� This is shown in the figure below:

Thus,

Construction� of angle 105o .

Angle 105o� can be constructed by constructing 60o in the adjacent right angle at E and bisecting the remaining 30o. Thus 105 = 90 + � x 30.this is shown in the� figure below:

Thus

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To construct angle 120o

the following steps must betaken :

1. Draw a straight line AB and mark a point C on the line where the angle 120o is to be constructed.
2. With centre C and a suitable radius, draw a well extended arc to cut line CB at point D.
3. With centre D and the same radius used in step 2 above draw an arc to cut the extended arc in step 2�

at point E.

4. With E as centre and the same radius, draw an arc to cut the extended arc at point F.
5. Draw line CF. Thus

Construction of angle 150 o.

Since 150 = 120o� + � of 60o, first construct angle 120o on� a straight line angle. Then bisect the adjacent 60o angle to get 30o.� Thus 30o + 120o on the right hand side gives the required angle 150o.� This is shown in the figure below:

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EVALUATION�

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• Construct angle 70o

• (a) Construct angle 105o

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(b) Construct angle 150o.

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GENERAL EVALUATION

Construct the following angles using compasses only.�

1. 60o
2. 45o
3. 135o
4. 37.5o

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READING ASSIGNMENT�

NGM SS Bk I pg 176-178;Ex.16a. No 1 pg 177

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WEEKEND ASSIGNMENT�

Construct ∆XYZ such that XY = 5cm, XYZ = 120o and YZ = 7cm. measure the following�

1. XZ��� ��� A. 10.4cm� � � B. 13cm��� � � C. 8cm��� D. 4cm
2. YXZ��� A. 25o �����B. 30o � � C. 35o� ������D. 40o
3. XZY��� A. 30o � � � B. 250� �����C. 50o � � � D. 60o

Construct ∆ABC such that AB = 6cm, BC = 7.5cm and ABC = 75o. bisect AB at P and AC at Q. Measure�

1. PQ��� ��� A.� 3.8cm� � � B. 10cm� � � C. 2cm� � � D. 8cm
2. QC��� ��� A. 5.1cm � � � B. 6.8cm��� � C. 4.1cm� � � D. 8.2cm

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THEORY

1. (a)Use ruler and compasses to construct� � � PQR in which� Q = 90o, /QR/ = 5cm and /PR/ = 10CM

(b) Measure /PQ/

(c) Use pytahgoras� theorem to check the result.

1. (a) Construct� � � ABC such that /AB/ = 7cm, /BC/ = 6cm and ABC = 60o

(b) The bisector of C meets the perpendicular bisector of AC at X. Find the point X by construction

(c) Measure !BX