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12.1 Recap: Angles and Scale Drawing
The concepts in this unit rely heavily on knowledge acquired previously, particularly for angles and scale drawings, so in this first section we revise these two topics.
Example 1
In the diagram opposite, determine the size of each of the unknown angles.
Solution
Since c + 100 ° = 180 ° (BCD is a straight line) c = 180 ° − 100 ° c = 80 ° Also, b = c , since the triangle is isosceles, so b = 80 °. Finally, since a + b + c = 180 °(angles in a triangle add up to 180 °) then a = 180 ° − ( 80 ° + 80 °) so a = (^20) °
Example 2
In the diagram opposite, given that a = 65 °, determine the size of each of the unknown angles.
Solution
b = 180 ° − a (angles on a straight line are supplementary, i.e. they add up to 180 °) b = 180 ° − 65 ° b = 115 ° c = a = 65 ° (vertically opposite angles) d = b = 115 °^ (corresponding angles, as the lines are parallel) e = a = 65 ° (corresponding angles) f = a = 65 ° (alternate angles)
12 Constructions and Loci
c^ 100˚ b
a
A
B C
D
(BCD is a straight line.)
d^ e
cb a
f
Example 3
Draw an accurate plan of the car park which is sketched here. Use the scale 1 cm ≡ 10 m. Estimate the distance AB.
Solution
The equivalent lengths are: 100 m ≡ 10 cm, 80 m ≡ 8 cm, 60 m ≡ 6 cm, giving the following scale drawing:
A
B
In the scale drawing, AB = 11.7 cm, which gives an actual distance AB = 117 m in the car park.
60 m
80 m
100 m
60 m
A
B
- (a) The time on this clock is 3 o'clock. What is the size of the angle between the hands?
(b) Write down the whole number missing from this sentence: At ......... o'clock the size of the angle between the hands is 180 °. (c) What is the size of the angle between the hands at 1 o'clock? (d) What is the size of the angle between the hands at 5 o'clock? (e) How long does it take for the minute hand to move 360 °? (KS3/99/Ma/Tier 3-5/P2)
- (a) Which two of these angles are the same size?
(b) Draw an angle which is bigger than a right angle. (c) Kelly is facing North. She turns clockwise through 2 right angles. Which direction is she facing now? (d) Aled is facing West. He turns clockwise through 3 right angles. Which direction is he facing now? (KS3/98/Ma/Tier 3-5/P1)
A B^ C
E
D
N
S
W E
- The shape below has 3 identical white tiles and 3 identical grey tiles. The sides of each tile are all the same length. Opposite sides of each tile are parallel. One of the angles is 70 °.
(a) Calculate the size of angle k.
(b) Calculate the size of angle m. Show your working. (KS3/99/Ma/Tier 4-6/P1)
- Kay is drawing shapes on her computer.
(a) She wants to draw this triangle. She needs to know angles a, b and c. Calculate angles a, b and c.
(b) Kay draws a rhombus: Calculate angles d and e.
(c) Kay types the instructions to draw a regular pentagon: repeat 5 [forward 10, left turn 72] Complete the following instructions to draw a regular hexagon: repeat 6 [forward 10, left turn .........] (KS3/97/Ma/Tier 4-6/P1)
c
a
40˚ b
6
80˚
NOT TO SCALE
NOT TO SCALE
50˚
10
10
10
10
e
d
k
70˚ m (^) NOT TO SCALE
- Look at the diagram: Side AB is the same length as side AC. Side BD is the same length as side BC. Calculate the value of x. Show your working.
(KS3/99/Ma/Tier 6-8/P1)
12.2 Constructions
In this section we look at how to construct triangles and various lines. You will need a ruler, a protractor and a pair of compasses to be able to draw these constructions. The following examples illustrate some of the techniques that you will need to use.
Example 1
Construct the perpendicular bisector of the line AB. Then label the midpoint of AB, M.
Solution
There are many lines that cut AB exactly in half. We have to construct the one that is perpendicular to AB. We begin by drawing arcs of equal radius, centred on the points A and B, as shown in the diagram. The radius of these arcs should be roughly 23 to 34 of the length AB.
Then draw a line through the intersection points of the two arcs. The point where the bisector intersects AB can then be labelled M.
A B
Perpendicular bisector
M
A B
B C
D
A
x ˚ 3 x ˚
NOT TO SCALE
A B
Example 2
The diagram shows the line AB and the point C. Draw a line through C that is perpendicular to AB.
Solution
Using C as the centre, draw an arc as shown.
Then using the intersection points of this arc with the line AB as centres, draw two further arcs with radii of equal length. The perpendicular line can then be drawn from C through the point where these two new arcs cross.
Example 3
Bisect this angle.
Solution
To bisect an angle you need to draw a line that cuts the angle in half. First draw an arc using O as the centre.
A B
C
O
O
A B
C
A B
C
Exercises
- (a) Draw a line of length 10 cm. (b) Construct the perpendicular bisector of the line. (c) Check that it does cut the line in half. (d) Use a protractor to check that it is perpendicular.
- (a) Mark 3 points, not in a straight line, on a piece of paper and label them A, B and C. Draw a line from A to B. (b) Construct a line that is perpendicular to AB and passes through C. (c) Use a protractor to check that your line is perpendicular.
- (a) Use a ruler and a protractor to construct the triangle ABC where AB = 6 cm, ∠ ABC= 60 °and ∠ BAC = 50 °. (b) Construct a line that is perpendicular to AC and passes through the corner B.
- (a) Draw a triangle with sides of length 7 cm, 4 cm and 6 cm. (b) Construct the perpendicular bisector of each side. What do you notice? (c) Draw a circle with its centre at the point where the lines intersect and that passes through each corner of the triangle. (d) Repeat this process for any other triangle. Does it still work?
- (a) Draw the triangle which has sides of length 8 cm, 7 cm and 6 cm. (b) Construct the bisector of each angle of the triangle. (c) Using the point where the lines intersect as its centre, draw the largest circle that will fit inside the triangle.
- The diagram shows how Ishmael constructed a 60 ° angle. (a) Construct a 60 ° angle in this way and then check that it is 60 °. (b) Bisect your angle to obtain a 30 ° angle. (c) Construct the following angles, using a pair of compasses and a ruler. (i) 120 ° (ii) 240 ° (iii) 300 ° (iv) (^90) ° (v) (^270) ° (vi) (^45) °
60˚
- The triangle ABC is such that AB^ =^ 6 cm, AC^ =^ 7 cmand ∠^ BAC^ =^50 °. (a) Draw the triangle. (b) What is the length of the side BC? (c) Construct a line that passes through C and is perpendicular to AB. (d) Hence calculate the area of the triangle.
- A triangle PQR has PR = 6 cm, QR = 5 cm and ∠ QPR= 45 °. Abigail and Kirsty are asked to draw this triangle. They draw the two triangles below.
(a) Are they both correct? (b) Draw the two possible triangles ABC, given the information below. AB = 8 cm BC = 7 cm ∠ BAC = 50 °
- Construct each of the following triangles, without using a protractor. (a) (b)
Abigail's Q Triangle
P R
Kirsty's Triangle
R P R
Q
30˚ 45˚ 7 cm 120˚
5 cm
6 cm
(KS3/96/Ma/Tier 5-7/P1)
- (a) The top and the base of this box are semi-circles. Which one of the nets below could fold up to make a box like this?
A B
C
D
E
(b) This is a rough sketch of the base of a box. It is a semi-circle, with diameter 8 cm. Make an accurate, full size drawing of the base of the box. You will need a ruler and a pair of compasses. (KS3/98/Ma/Tier 3-5/P2)
8 cm
Example 3
Draw the locus of points that are 1 cm from this circle.
Solution
The locus is made up of 2 parts. 1 part consists of the points that are 1 cm from the circle and inside it; the other is those points that are 1 cm from the circle and are outside it.
Exercises
- (a) Draw a line of length 5 cm. (b) Draw the locus of points that are 1 cm from the line.
- (a) Draw a circle of radius 2 cm. (b) Draw the locus of points that are 2 cm from the circle. (c) On your diagram, shade the locus of points that are less than 2 cm from the circle.
- (a) Draw the rectangle shown in the diagram. (b) Draw the locus of the points that are 1 cm from the rectangle. (c) Repeat part (b) for a rectangle that is 6 cm long and 5 cm wide.
3 cm
4 cm 1 cm
- Construct the locus of the points that are equidistant from the two lines shown in the diagram.
- (a) Construct the triangle shown in the diagram.
(b) Draw the locus of the points that are 1 cm from the triangle.
- Draw the locus of the points that are 1 cm from the shape in the diagram.
- Two points A and B are 6 cm apart. (a) Draw the locus of the points that are equidistant from A and B. (b) Draw the locus of points that are 5 cm from B. (c) Indicate the points that are 5 cm from A and B.
- The points A and B are 9 cm apart. Draw the locus of the points that are twice as far from A as they are from B.
- (a) Construct the triangle shown in the diagram.
(b) Draw the locus of points that are equidistant from A and B and within 3 cm of C.
4 cm
6 cm
4 cm
4 cm
6 cm
A
B
C
4 cm
3 cm (^) 6 cm
