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MEP Pupil Text 4
4 Trigonometry
4.1 Squares an and Tr Triangles A triangle is a geometric shape with with three sides and three three angles. Some of the different different types of triangles are described in this Unit. A square is a four-sided geometric shape with with all sides of equal length. All four angles are the same size.
An isosceles triangle has two sides that are the same length, and the two base angles are equal.
60˚
All the sides of an equilateral triangle are the same length, and the angles are all 60° . 60˚
60˚
A scalene triangle has sides that all have different lengths, and has 3 different angles.
Worked Example 1 the area of the square in the diagram. Master yourFindsemester withshown Scribd & The New Solution York Times Special offer for students: Only $4.99/month.
Area
=
4
×
4
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4 cm
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MEP Pupil Text 4
(c )
2 cm
(d)
2 cm
2 cm
40˚
40˚
(e )
30˚
(f) 60˚
60˚
60˚
(g)
60˚
(h) 3 cm 5 cm
3 cm 7 0˚
2.
3.
7 0˚
(a)) (a
When a squ When squar aree is cu cutt in in ha half lf di diag agon onal ally ly,, two two tr tria iang ngle less ar aree ob obta tain ined ed.. Ar Aree these triangles scalene, isosceles or equilateral?
(b)
Whatt type Wha type of trian triangle gle do do you get get if you you cut cut a rectan rectangle gle in in half half diagon diagonally ally??
Fin Fi nd the the ar area ea of ea each ch sq squa uare re bel elo ow. (a )
(b) 5 cm
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7 cm
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1 cm
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MEP Pupil Text 4
(c )
2 cm
(d)
2 cm
2 cm
40˚
40˚
(e )
30˚
(f) 60˚
60˚
60˚
(g)
60˚
(h) 3 cm 5 cm
3 cm 7 0˚
2.
3.
7 0˚
(a)) (a
When a squ When squar aree is cu cutt in in ha half lf di diag agon onal ally ly,, two two tr tria iang ngle less ar aree ob obta tain ined ed.. Ar Aree these triangles scalene, isosceles or equilateral?
(b)
Whatt type Wha type of trian triangle gle do do you get get if you you cut cut a rectan rectangle gle in in half half diagon diagonally ally??
Fin Fi nd the the ar area ea of ea each ch sq squa uare re bel elo ow. (a )
(b) 5 cm
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7 cm
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1 cm
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MEP Pupil Text 4
6.
Two squ square aress of of side side 4 cm are joi joined ned tog togeth ether er to to form form a recta rectangl ngle. e. Wha Whatt is the are areaa of the rectangle?
7.
A squ square are of sid sidee 12 12 cm is cut cut in in half half to form form two tri triang angles les.. Wha Whatt is the are areaa of of each each triangle?
8.
A squ square are of sid sidee 6 cm is is cut cut into into qua quarte rters rs to to form form 4 smal smaller ler squ square ares. s. Wha Whatt is the area of each of these squares?
4.2 Pythagoras' Theorem Pythagoras' Theorem gives a relationship between the lengths of the sides of a right angled triangle. For the triangle shown, shown, a 2 = b2 + c2
a
b
Note
c
The longest side of a right angled triangle is called the hypotenuse.
Worked Example 1
4 cm
Find the length of the hypotenuse of the triangle shown in the diagram. Give your answer answer correct to 2 decimal places.
6 cm
Solution As this is a right angled triangle, Pythagoras' Pythagoras' Theorem can be used. used. If the length of the hypotenuse is a, then b = 4 and c = 6 .
+ c a = b with Master yourSosemester Scribd a =4 +6 & The New York Times a = 16 + 36 2
2
2
2
2
2
2
Special offer for students: Only $4.99/month. a 2 = 52
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MEP Pupil Text 4
So
a
2
=
+
c
6
2
= x +
2
3
36
= x +
2
9
36
−
9
=
x2
36
−
9
=
x2
b
2
2 2
27 = x
2
27
=
x
x
=
5.2 cm
(to one decimal place)
Exercises 1.
Fin Fi nd the the le leng ngth th of th thee sid sidee mar marke ked d x in each triangle. (a )
(b) x
4m
5 cm
x
12 cm 3m
(c )
(d) 26 m
15 cm
10 m
x
x 12 cm
6 cm
x
(e ) 6m
(f) 8m
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(g)
x
10 cm
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MEP Pupil Text 4
2.
Find the length of the side marked x in each triangle. Give your answers correct to 2 decimal places. (a)
(b) x
7 cm
x
15 cm 11 cm 14 cm
4 cm
(c)
7m
(d) 5m
8 cm
x
(e)
x
8 cm
(f) 10 cm
x
x
12 cm
7 cmYou're Reading a Preview Unlock full access with a free trial.
5m
(g) 6m
Download With Free Trial
x
x
Master your semester with Scribd & The New York Times 18 cm 10 cm (i)
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2m
(h)
12 m
x
(j)
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5m
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MEP Pupil Text 4
(m)
(n)
x
8.9 cm
5.2 cm x
3.
5.4 m
2.3 m
Ben
Adam runs diagonally across a school field, while Ben runs around the edge.
Adam 200 m
(a)
How far does Ben run?
(b)
How far does Adam run?
(c)
How much further does Ben run than Adam? 120 m
4.
A guy rope is attached to the top of a tent Guy pole, at a height of 1.5 metres above the rope ground, and to a tent peg 2 metres from the You're a Preview Peg base of the pole. How long is theReading guy rope?
Pole 1.5 m
2m
Unlock full access with a free trial.
5.
Farida is 1.4 metres tall. At a certain time her shadow is 2 metres long. What is the Download With Trial distance from the top of her head to the topFree of her shadow?
6.
A rope of length 10 metres is stretched from the top of a pole 3 metres high until it reaches ground level. How far is the end of the line from the base of the pole?
A rope is fixed between two trees that are Master your7. semester with Scribd 10 metres apart. When a child hangs on to the centre of the rope, it sags so that the & The New York Times centre is 2 metres below the level of the Special offer for students: Only $4.99/month.
ends. Find the length of the rope.
2m
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MEP Pupil Text 4
9.
A
The picture shows a garden shed. Find the length, AB, of the roof.
B 2.5 m 2m
10.
2m
Miles walks 3 km east and then 10 km north. (a)
How far is he from his starting point?
(b)
He then walks east until he is 20 km from his starting point. How much further east has he walked?
11.
Ali is building a shed. It should be rectangular with sides of length 3 metres and 6 metres. He measures the diagonal of the base of the shed before he starts to put up the walls. How long should the diagonal be?
12.
Pauline is building a greenhouse. The base PQRS of the greenhouse should be a 1.4 m rectangle measuring 2.6 metres by 1.4 metres. You're Reading a Preview
P
To check the base is rectangular Pauline has Unlock full access with a free trial. to measure the diagonal PR.
S
2.6 m
(a)
Calculate the length of PR when theFree baseTrial is rectangular. You must show all Download With your working.
(b)
When building the greenhouse Pauline finds angle PSR > 90° . She measures PR. Which of the following statements is true? X:
PR is greater than it should be.
Y:
PR is less than it should be.
Z: PR is the right length. Master your semester with Scribd & The New York Times Special offer for students: Only $4.99/month.
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(SEG)
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MEP Pupil Text 4
Solution First consider the lower triangle. The unknown length of the hypotenuse has been marked y. y 2
=
42 + 4 2
y 2
=
16
y 2
=
32 cm
+
y
4 cm
16 4 cm
2 The upper triangle can now be considered, using the value for y .
From the triangle, x 2 = y 2 + 2 2 , and using y 2 = 32 gives x 2
=
32
x 2
=
36
x
=
36
x
=
6 cm
+
2 cm
4 x y
Note 2 32 , but to simply use y = 32 .
When finding the side x, it is not necessary to find
Worked Example 2
You're Reading a Preview Unlock full access with a free trial.
Find the value of x as shown on the diagram, and state the lengths of the two unknown sides.
13 m
Download With Free Trial
2 x
Solution Using Pythagoras' Theorem gives
3 x 2
132
=
( 2 x )
169
=
4 x 2 + 9 x 2
169
=
13 x 2
+
Master your semester with Scribd 13 = x x = 13 & The New York Times 2
Special offer for students: Only $4.99/month.
=
3.61 cm
(3 x )
2
(since (2 x )
2
=
22 x2 = 4 x 2 )
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(to 2 decimal places)
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MEP Pupil Text 4
(c)
x
(d)
x
4 10
6
12
4 3
(e)
(f) 4
10 15
x
2
x
2
2.
3
Find the length of the side marked x in the following situations. (a)
(b)
x
20
8
4 x
You're Reading a Preview x
Unlock full access with a free trial.
(c)
(d)
4 x
x
x
5 x
Download With Free Trial 4 x 20
5
3.
Which of the following triangles are right angled triangles? (a) (b) (c) (d)
9 Master your semester with Scribd B & The New York Times 10 13 26
Special offer for students: Only $4.99/month. A
15
Read Free Foron 30this Days Sign up to vote title 6.5 Useful Not useful 130 Cancel anytime. 6 120 C D
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MEP Pupil Text 4
5.
The diagram shows how the sign that hangs over a Fish and Chip shop is suspended by a rope and a triangular metal bracket. Find the length of the rope.
Rope 160 cm
50 cm 90 cm
6.
The diagram shows how a cable is attached to the mast of a sailing dingy. A bar pushes the cable out away from the mast. Find the total length of the cable.
Metal Bracket
FISH AND CHIPS
Wall
Mast 80 cm Bar 40 cm
You're Reading a Preview 240 cm
Unlock full access with a free trial.
Cable
Download With Free Trial
80 cm
7.
Deck
A helicopter flies in a straight line until it reaches a point 20 km east and 15 km north of its starting point. It then turns through 90 ° and travels a further 10 km.
Master your semester with Scribd (a) How far is the helicopter from its startingRead point? Free Foron 30this Days Sign up to vote title & The New York Times Useful Not useful how far would it end up from its (b) If the helicopter turned 90 ° the other way, Special offer for students: Only $4.99/month. starting point?
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MEP Pupil Text 4
9.
10.
A simple crane is to be constructed using an isosceles triangular metal frame. The top of the frame is to be 10 metres above ground level and 5 metres away from the base of the crane, as shown in the diagram. Find the length of each side of the triangle.
10 m
5m
A thin steel tower is supported on one side by two cables. Find the height of the tower and the length of the longer cable.
10 m 40 m
25 m
11.
An isosceles triangle has two sides of length 8 cm and one of length 4 cm. Find the height of the triangle and its area.
height
You're Reading a Preview Unlock full access with a free trial.
12.
Find the area of each the equilateral triangles that have sides of lengths (a)
8 cm
Download With Free Trial (b)
20 cm
(c)
2 cm
4.4 Sine, Cosine and Tangent When working in a right angled triangle, the longest side is known as the hypotenuse. The other two sides are known as the opposite and the adjacent. The adjacent is the side next to a marked angle, and the opposite side is opposite Special offer for students: $4.99/month. thisOnly angle.
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