CHAPTER - Overton Grange Maths KS4 - home CHAPTER 31 Pythagoras’ theorem31 CHAPTER Pythagoras’...
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562 31 CHAPTER Pythagoras’ theorem 31.1 Pythagoras’ theorem Pythagoras was a famous mathematician in Ancient Greece. The theorem which is named after him is an important result about right-angled triangles. Here is a right-angled triangle ABC. The angle at C is the right angle. The side, AB, opposite the right angle is called the hypotenuse. It is the longest side in the triangle. The right-angled triangle in the diagram has sides of length 3 cm, 4 cm and 5 cm. Squares have been drawn on each side of the triangle and each square has been divided up into squares of side 1 cm. The area of the square on the side of length 3 cm is 9 cm 2 . The area of the square on the side of length 4 cm is 16 cm 2 . The area of the square on the side of length 5 cm (the hypotenuse) is 25 cm 2 . Notice that 25 9 16, that is 5 2 3 2 4 2 In other words, 5 2 (the area of the square on the hypotenuse) is equal to the sum of 3 2 and 4 2 (the areas of the squares on the other two sides added together). This is an example of Pythagoras’ theorem, which is only true for right-angled triangles. Area of square R Area of square P Area of square Q Pythagoras’ theorem is used to find the length of the third side of a right-angled triangle, when the lengths of the other two sides are known. For this, the theorem is usually stated in terms of the lengths of the sides of the triangle. That is a b c c 2 a 2 b 2 Pythagoras’ theorem states: In a right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. A C B R P Q 25 9 16 4 3 5
Transcript of CHAPTER - Overton Grange Maths KS4 - home CHAPTER 31 Pythagoras’ theorem31 CHAPTER Pythagoras’...
562
CHAPTER 31 Pythagoras’ theorem
31C H A P T E R
Pythagoras’ theorem
31.1 Pythagoras’ theoremPythagoras was a famous mathematician in Ancient Greece.The theorem which is named after him is an important result about right-angled triangles.
Here is a right-angled triangle ABC.
The angle at C is the right angle.The side, AB, opposite the right angle is called the hypotenuse.It is the longest side in the triangle.
The right-angled triangle in the diagram has sides of length 3 cm, 4 cm and 5 cm.
Squares have been drawn on each side of the triangle and each square has been divided up into squares of side 1 cm.
The area of the square on the side of length 3 cm is 9 cm2.The area of the square on the side of length 4 cm is 16 cm2.The area of the square on the side of length 5 cm (the hypotenuse) is 25 cm2.
Notice that 25 � 9 � 16, that is 52 � 32 � 42
In other words, 52 (the area of the square on the hypotenuse) is equal to the sum of 32 and42 (the areas of the squares on the other two sides added together).
This is an example of Pythagoras’ theorem, which is only true for right-angled triangles.
Area of square R � Area of square P � Area of square Q
Pythagoras’ theorem is used to find the length of the third side of a right-angled triangle,when the lengths of the other two sides are known. For this, the theorem is usually statedin terms of the lengths of the sides of the triangle.
That is
a
b cc2 � a2 � b2
Pythagoras’ theorem states:In a right-angled triangle, the area of the squareon the hypotenuse is equal to the sum of theareas of the squares on the other two sides.
A
C B
R
P
Q
25
9
16 4
3
5
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In triangle DEF, Pythagoras’ theorem gives
DE2 � EF2 � DF2
(DE2 means that the length of the side DE is squared.)
31.2 Finding the length of a hypotenuseUse Pythagoras’ theorem to work out the length of the hypotenuse of a right-angledtriangle when the lengths of the two shorter sides are given.
Work out the length of the hypotenuse in this triangle
Solution 1c2 � a2 � b2
c2 � 82 � 152
c2 � 64 � 225
c2 � 289
c � �289� � 17
Length of hypotenuse � 17 cm
Work out the length of the side marked x in this triangle.Give your answer correct to 1 decimal place.
Solution 2c2 � a2 � b2
x2 � 5.32 � 12.42
x2 � 28.09 � 153.76x2 � 181.85
x � �181.85�� 13.48…
x � 13.5 cm (to 3 s.f.)
It is important to be able to apply Pythagoras’ theorem when the triangle is in a different position.
Give the final answer correct to 1 decimal place.
Use a calculator to find the square root. Write down at least four figures.
Work out 5.32 and 12.42 and add the results.
Substitute the given lengths.
State Pythagoras’ theorem.
Example 2
The answer is sensible, because the hypotenuseis longer than the other two sides.
Find �289� on a calculator.
c is a number which, when squared, gives 289In other words, c is the square root of 289
Work out 82 and 152 and add the results.
Substitute the given lengths.
State Pythagoras’ theorem.
Example 1
8 cm
15 cm c cm
12.4 cm
5.3 cmx
D
F E
31.2 Finding the length of a hypotenuse CHAPTER 31
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CHAPTER 31 Pythagoras’ theorem
In triangle XYZ, angle X � 90°, XY � 8.6 cm and XZ � 13.9 cm. Work out thelength of YZ.
Give your answer correct to 1 decimal place.
Solution 3
YZ2 � XY2 � XZ2
YZ2 � 8.62 � 13.92
YZ2 � 73.96 � 193.21
YZ2 � 267.17
YZ � �267.17� = 16.34…
YZ � 16.3 cm (to 3 s.f.)
Exercise 31A
1 Work out the length of the sides marked with letters in these triangles.
a b c
2 Work out the length of the sides marked with letters in these triangles.Give each answer correct to 1 decimal place.
a b c d
4.8 cm
10.6 cm
d6.2 cm
8.3 cm
c4.8 cm
9.1 cm
b
7.9 cm
7.4 cm
a
8 cm
6 cm
c40 cm
9 cmb
12 cm
5 cm
a
Give the final answer correct to 1 decimal place.
Use a calculator to find the square root. Write down at least four figures.
Work out 8.62 and 13.92 and add the results.
Substitute the given lengths.
State Pythagoras’ theorem.
The hypotenuse is the side opposite the right angle. Angle X is the right angle so the hypotenuse is YZ.
13.9 cm
8.6 cm
X
Y
Z
Example 3
3 a In triangle ABC, angle A � 90°, AB � 3.4 cm and AC � 12.1 cm.Work out the length of BC.Give your answer correct to 1 decimal place.
b In triangle DEF, angle E � 90°,DE � 6.3 cm and EF � 9.8 cm.Work out the length of DF.Give your answer correct to 1 decimal place.
c In triangle PQR, angle R � 90°,PR � 5.9 cm and QR � 13.1 cm.Work out the length of PQ.Give your answer correct to 1 decimal place.
d In triangle XYZ, angle X � 90°,XY � 12.6 cm and XZ � 16.5 cm.Work out the length of YZ.Give your answer correct to 1 decimal place.
31.3 Finding the length of one of the shorter sides of a right-angled triangle
So far, the lengths of the two shorter sides of a right-angled triangle have been given andthe length of the hypotenuse has been worked out.
Pythagoras’ theorem can also be used to work out the length of one of the shorter sides ina right-angled triangle when the lengths of the other two sides are known.
Work out the length of the side marked a in this triangle.Give your answer correct to 2 decimal places.
Solution 4c2 � a2 � b2
14.12 � 10.22 � a2
198.81 � 104.04 � a2
198.81 � 104.04 � a2
94.77 � a2
Subtract 104.04 from both sides.
Work out 14.12 and 10.22.
Substitute the given lengths.
State Pythagoras’ theorem.
Example 4
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31.3 Finding the length of one of the shorter sides of a right-angled triangle CHAPTER 31
6.3 cm
9.8 cmE F
D
16.5 cm
12.6 cm
Z
Y
X
3.4 cm
12.1 cmA C
B
13.1 cm
5.9 cm
R
P
Q
10.2 cm14.1 cm
a
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CHAPTER 31 Pythagoras’ theorem
or a2 � 94.77
a � �94.77� = 9.734…
a � 9.73 cm
In triangle ABC, angle A � 90°, BC � 17.4 cm and AC � 5.8 cm.Work out the length of AB.Give your answer correct to 1 decimal place.
Solution 5
BC2 � AC2 � AB2
17.42 � 5.82 � AB2
302.76 � 33.64 � AB2
302.76 � 33.64 � AB2
269.12 � AB2
AB � �269.12�� 16.40…
AB � 16.4 cm (to 3 s.f.)
Exercise 31B
1 Work out the lengths of the sides marked with letters in these triangles.
a b c d
2 Work out the lengths of the sides marked with letters in these triangles.Give each answer correct to 2 decimal places.a c
b
d
20 cm
25 cmc
12 cm37 cm
b24 cm
25 cma
Give the final answer correct to 1 decimal place.
Use a calculator to find the square root. Write down at least four figures.
Subtract 33.64 from both sides.
Work out 17.42 and 5.82.
Substitute the given lengths.
State Pythagoras’ theorem.
Angle A is the right angle so the hypotenuse is BC.
Example 5
9.73 cm is less than the length of the hypotenuse14.1 cm and so the answer is sensible.
Give the final answer correct to 2 decimal places.
Use a calculator to find the square root. Write down at least four figures.
17.4 cm
5.8 cm
B
A C
2.5 cm
6.5 cm
d
8.3 cm
2.1 cm
d12.4 cm
1.8 cm
c
11.3 cm
8.1 cm
b
4.8 cm10.7 cm
a
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31.4 Applying Pythagoras’ theorem CHAPTER 31
3 a In triangle ABC, angle A � 90°, AB � 5.9 cm and BC � 16.3 cm.Work out the length of AC. Give your answercorrect to 1 decimal place.
b In triangle DEF, angle E � 90°, DF � 10.1 cm and EF � 7.8 cm.
i Draw a sketch of the right-angled triangle DEF andlabel sides DF and EF with their lengths.
ii Work out the length of DE. Give your answer correct to 2 decimal places.
31.4 Applying Pythagoras’ theoremPythagoras’ theorem can be used to solve problems.
A boat travels due North for 5.7 km. The boat then turns and travels due East for 7.2 km.Work out the distance between the boat’s finishing point and its starting point.
Give your answer correct to 2 decimal places.
Solution 6
d2 � 5.72 � 7.22
d2 � 32.49 � 51.84d2 � 84.33d � �84.33�� 9.183…
Distance � 9.18 km (to 2 decimal places)
Isosceles triangles can be split into two right-angled triangles, in which Pythagoras’theorem can be used.
The diagram shows an isosceles triangle ABC.The midpoint of BC is the point M.In the triangle, AB � AC � 8 cm and BC � 6 cm.
a Work out the height, AM, of the triangle.Give your answer correct to 2 decimal places.
b Work out the area of triangle ABC. Give your answer correct to 1 decimal place.
Example 7
Draw a sketchof the boat’sjourney.
5.7 km
Start
Finish7.2 km
d km
Example 6
5.9 cm16.3 cm
B
A C
The sketch is of a right-angled triangle sothat Pythagoras’ theorem can be used.
The distance between the starting point and thefinishing point is the length of the hypotenuse of thetriangle. This distance is marked d km in the sketch.
Remember that the points of the compass are
N
S
W E
8 cm8 cm
A
CB
6 cmM
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CHAPTER 31 Pythagoras’ theorem
Solution 7Pythagoras’ theorem cannot be used in triangle ABC as this triangle is not right-angled.
a
By Pythagoras
AB2 � AM2 � BM2
82 � h2 � 32
64 � h2 � 964 � 9 � h2
55 � h2
h � �55� � 7.416…h � 7.42
Height of triangle � 7.42 cm (to 2 d.p.)
b area � �12� � 6 � 7.416…
area � 22.24…
area � 22.2 cm2 (to 1 d.p.)
Length of a line joining two points
The diagram shows the points A(1, 1) and B(9, 5).The right-angled triangle ABC has been drawn so that AC � 8 and BC � 4
Pythagoras’ theorem can be used to find the length of AB.AB2 � 82 � 42
AB2 � 64 � 16 � 80AB � �80� � 8.94 (to 2 d.p.)
Find the length of the line joining A(3, 2) and B(15, 7)
Solution 8
Draw a sketch showing A and B and complete theright-angled triangle ABC.AC � 15 � 3 � 12BC � 7 � 2 � 5
3 15
212
5
7
y
A
B
C
O x
Example 8
For triangle ABC base � 6 cm andheight � 7.416… cm.
area of a triangle � �12� � base � height
Triangle ABM is right angled with hypotenuse AB. Theheight, AM, of the triangle is marked h cm on the sketch.
Draw a sketch of triangle ABM.
BM � 3 cm as M is the midpoint of BC.
As M is the midpoint of the base of the isosceles triangle,the line AM is the line of symmetry of triangle ABC. SoAM is perpendicular to the base and angle AMB � 90°.
3 cm
8 cmh cm
B M
A
1
1 2 3 4 5 6 7 8
8
4
9 10
2
3
4
5
6
y
O
A C
B
x
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31.4 Applying Pythagoras’ theorem CHAPTER 31
AB2 � 122 � 52 AB2 � 144 � 25 � 169AB � �169� � 13
Exercise 31C
1 Find the lengths of the sides marked with letters in each of these triangles. Give eachanswer correct to 1 decimal place.
a b c
d e f
2 The diagram shows a ladder leaning against a vertical wall.The foot of the ladder is on horizontal ground. The length of theladder is 5 m. The foot of the ladder is 3.6 m from the wall.Work out how far up the wall the ladder reaches.Give your answer correct to 2 decimal places.
3 Aiton (A), Beeville (B) and Ceaborough (C) are three towns as shown in this diagram.Beeville is 10 km due South of Aiton and 21 km due East of Ceaborough. Work out the distance between Aitonand Ceaborough. Give your answer correct to the nearest km.
4 a Work out the length of the side marked b cm in the triangle. Give your answer correct to 2 decimal places.
b Work out the area of the triangle.Give your answer correct to 2 decimal places.
5 a Work out the length of the side marked a cm in the triangle. Give your answer correct to 2 decimal places.
b Work out the perimeter of the triangle.Give your answer correct to 1 decimal place.
6 The diagram represents the end view of a tent,triangle ABC; two guy-ropes, AP and AQ; and avertical tent pole, AN. The tent is on horizontalground so that PBNCQ is a straight horizontal line.Triangles ABC and APQ are both isosceles triangles.
BN � NC � 2 m, AN � 2.5 m and AP � AQ � 5 m
A
P B N C Q2 m
5 m2.5 m
11.7 cm
8.3 cm
a cm
10.6 cm14.5 cm
b cm
21 km
10 km
A
BC
3.6 m
5 m
8.4 cm
8.4 cm
f
9.9 cm
13.8 cm e
11.4 cm
14.8 cmd
6.8 cm
9.7 cmc
18.6 mm8.2 mm
b
6.8 cm
13.4 cm
a
Use Pythagoras’ theorem to find the length of AB.
a Work out the length of the side AC of the tent. Give your answer correct to 2decimal places.
b Work out the length of i NQ, ii CQ.Give your answers correct to 2 decimal places.
There is a tent peg at P and a tent peg at Q.
c Work out the distance between the two tent pegs at P and Q. Give your answercorrect to 2 decimal places.
7 The diagram shows two right-angled triangles.
a Work out the length of the side marked x.
b Hence work out the length of the side marked y.
8 Work out the length of the line joining each of these pairs of points.
a (1, 3) and (5, 6) b (3, 1) and (11, 7) c (2, 1) and (14, 6)
9 Work out the length of the line joining each of these pairs of points. Give youranswers correct to 1 decimal place.
a (3, 5) and (4, 7) b (4, 2) and (7, 5) c (4, 5) and (0, 7)
Chapter summary
x
y
39 cm
12 cm
9 cm
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CHAPTER 31 Pythagoras’ theorem
You should now knowIn a right-angled triangle, the side opposite the right angle is the hypotenuse. It isthe longest side in the triangle.
You should now be able toUse Pythagoras’ theorem in right-angled triangles
● to find the length of the hypotenuse when the lengths of the other two sides areknown
● to find the length of one of the shorter sides of the triangle when the lengths ofthe other two sides are known
Apply Pythagoras’ theorem in problems involving right-angled triangles, including thelength of a line joining two points whose coordinates are given.
c2 � a2 � b2a
b
c
�
�
�
Chapter 31 review questions1 ABC is a right-angled triangle.
AB � 4 cm, BC � 6 cm.Calculate the length of AC.Give your answer in centimetres,correct to 2 decimal places. (1385 June 2001)
2 ABC is a right-angled triangle.AB � 8 cm, BC � 11 cm.Calculate the length of AC.Give your answer correct to 1 decimal place. (1388 Mar 2003)
3 ABC is a right-angled triangle.AC � 5 m, CB � 8.5 m.
a Work out the area of the triangle.
b Work out the length of AB.
Give your answer correct to 2 decimal places. (1385 Nov 2002)
4 The diagram shows triangle ABCAC � 16.4 cm, BC � 12.6 cm,Angle ABC � 90°.
Work out the length of AB.
Give your answer correct to 1 decimal place.
5 Calculate the length of AB.Give your answer correct to 1 decimal place.
(November 1997)
6 AC � 12.6 cm,BC � 4.7 cm,Angle ABC � 90°.
Calculate the length of AB.Give your answer correct to 1 decimal place. (1388 Jan 2004)
7 Angle MLN � 90°,LM � 3.7 m,MN � 6.3 m.
Work out the length of LN.Give your answer correct to 2 decimal places. (1388 March 2004)
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Chapter 31 Review questions CHAPTER 31
C
BA
18 cm12 cm
Diagram NOTaccurately drawn
C
BA
4.7 cm12.6 cm
Diagram NOTaccurately drawn
M N
L
6.3 m
3.7 m
Diagram NOTaccurately drawn
CB
A
12.6 cm
16.4 cm
Diagram NOTaccurately drawn
C B
A
5 m
8.5m
Diagram NOTaccurately drawn
B C
A
8 cm
11 cm
Diagram NOTaccurately drawn
B C
A
4 cm
6 cm
Diagram NOTaccurately drawn
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CHAPTER 31 Pythagoras’ theorem
8 Work out the length, in centimetres, of AM.Give your answer correct to 2 decimal places.
(1388 Mar 2003)
9 Ballymena is due West of Larne. Woodburn is 15 km due South of Larne. Ballymena is 32 km from Woodburn.
Calculate the distance of Larne from Ballymena. Give your answer in kilometres,correct to 1 decimal place. (1385 June 1998)
10 Sidney places the foot of his ladder on horizontal ground and the top against a vertical wall.The ladder is 16 feet long. The foot of the ladder is 4 feet from the base of the wall.
Work out how high up the wall the ladder reaches.Give your answer correct to 1 decimal place.
(1384 June 1995)
11 Calculate the length of a diagonal of this rectangle.Give your answer in centimetres correct to 1 decimal place. (1384 June 1995)
12 A, B, C and D are four points on the circumferenceof a circle.ABCD is a square with sides 20 cm long.
Work out the diameter, BD, of the circle.Give your answer correct to 1 decimal place. (1385 Nov 1998)
13 The diagram shows a rectangle drawn inside a circle. The centre of the circle is at O.
The rectangle is 15 cm long and 9 cm wide.
Calculate the circumference of the circle.Give your answer correct to 1 decimal place. (1385 Nov 2001)
14 A sheet of A5 paper is a rectangle 210 mm long and 148 mm wide.
a Calculate the area of a sheet of A5 paper.Give your answer in square centimetres.
b Calculate the length of a diagonal of the rectangle.Give your answer correct to the nearest millimetre. (1385 June 2002)
4 feet
16 feet
A B
D C
Diagram NOTaccurately drawn
O Diagram NOTaccurately drawn9 cm
15 cm
Diagram NOTaccurately drawn
148 mm
210 mm
15 cm
12 cm
M
A
B C
7 cm7 cm
Diagram NOTaccurately drawn
8cm
15 km32 km
Ballymena Larne
Woodburn
Diagram NOTaccurately drawn
N
15 ABC is a rectangle.AC � 17 cm. AD � 10 cm.
Calculate the length of the side CD.Give your answer correct to 1 decimal place. (1387 Nov 2004)
16 The diagram shows triangle ABC and circle, centre OA, B and C are points on the circumference of the circle.AB is a diameter of the circle.
AC � 16 cm and BC � 12 cm and angle ACB � 90°.
a Work out the diameter AB of the circle.
b Work out the area of the circle.Give your answer to the nearest cm2. (1387 June 2005)
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Chapter 31 Review questions CHAPTER 31
A
D
B
C
17 cm Diagram NOTaccurately drawn10 cm
A
O
BC
Diagram NOTaccurately drawn
12 cm
16 cm
