N5 - Chapter 4 this is page 26 Pythagoras

Pythagoras was a famous Greek Mathematician who discovered an amazing connection between the three sides of any right angled triangle.

He discovered that the 3 sides of a right angled triangle were connected mathematically by the formula :- b

a

c

Chapter 4Chapter 4 Pythagoras Pythagoras

Revision of Pythagoras’ Theorem

Calculating the length of the HYPOTENUSE of a right angled triangle

(You will have covered Pythagoras’ Theorem at an earlier stage.)

Example :- Calculate the length of the hypotenuse of this right angled triangle.

8 cm

11 cm

x cmSolution :- Begin with :- x 2 = 112 + 8 2

=> x 2 = 121 + 64 => x 2 = 185 => x = 185 = 13·6.

Calculating the length of one of the SMALLER sides in a R.A.T.

Example :- Calculate the length of the side marked a in this right angled triangle.

9 cm18 cm

a cm

Solution :- Begin with :- a2 = 182 – 92 => a2 = 324 – 81 => a2 = 243 => a = 243 = 15·6.

*note

Exercise 4·1

c 2 = a 2 + b 2

1. Calculate the lengths of the missing sides in thefollowing right angled triangles :-(a) (b)

(c) (d)8 cm

x cm8 cm

19 cm

15 cm

y cm

3·9 mm 4·1 mm

w mm

7·6 m

9·8 mz m

2. Shown is an isosceles triangle.

(a) Calculate the heightof the triangle.

(b) Now calculate its area.

3. Calculate the area of this rectangle :-

20 cm

26 cm

72 cm

75 cm

N5 - Chapter 4 this is page 27 Pythagoras

4. Calculate the perimeter of this right angled triangle :-

5. Calculate the value of x, which indicates the length of the slopingside of this trapezium.

6. This shape consists of a rectangle with an isosceles triangle attached to its end.

(a) Calculate the total length (L) of the figure.

(b) Now calculate its area.

7. Shown are the points P(–3, –4) and G(6, 3).

Draw a coordinate diagram, plot the two points and calculate the length of the line PQ.

8. Draw a new set of axes, plot the 2 pointsA(–1, 8) and B(6, –4) and calculate the length of the line AB.

15 cm

8 cm

6·4 m

9·7 m

14·5 m

x m

9. Shown is an isosceles triangular prism.

(a) Calculate the height of the triangle, represented by the dotted line.

(b) Calculate the volume of the prism.

10. When a boy was asked to calculate the value of x, he proceeded as follows :–

Explain in words, when the boy looked at his answer and at the triangle, why he should have known that his answer had to be wrong.

11. Shown is a trapezium with a line of symmetry.

Calculate the perimeter of the above trapezium.

12. A fortune teller has a “magic” glass globe.It rests embedded in a wooden plinth as shown.The plinth measures 32 cm by 7 cm.The diameter of the globe is 30 cm.

Calculate the overal height, h of the figure.

20 cm

32 cm

24 cm

L cm

x

y

•

•

P

Q

5 cm

6·5 cm

4 cm

x 2 = 10 2 + 7 2

=> x 2 = 100 + 49

=> x 2 = 149

=> x = √149 = 12·2 cm

x cm10 cm

7 cm

36 mm

96 mm

42 mm

30 cm

7 cm

24 cm

• h cm

N5 - Chapter 4 this is page 28 Pythagoras

1. Check if this triangle is right angled at Q.

Copy and complete :-

• PQ 2 = 18 2 = 324, • QR 2 = 7·5 2 = .....

• PR 2 = ....2 = ....

• PQ 2 + QR 2 = 324 + .... = ...... = PR2

• by the Converse of Pythagoras’ Theorem, triangle PQR must be r....... a...... at Q

2. Show that this triangle is NOT right angled.

i.e. (Show that UW 2 + VW 2 ≠ UV 2)

The CONVERSE of Pythagoras’ Theorem

Pythagoras’ Theorem only works on a right angled triangle.

We can use Pythagoras Theorem “in reverse” to actually prove that a triangle is right angled.

Example :-

Look at triangle ABC opposite

We can prove it is right angled as follows :-

• Write down the 3 sides :- AB = 5·2, AC = 3·9, BC = 6·5.

• Square each side :- AB 2 = 27·04, AC 2 = 15·21, BC 2 = 42·25.

• Add the two smaller squares together :- AB 2 + AC 2 = 27·04 + 15·21 = 42·25.

• Check if this is the same value as the largest square :- AB 2 + AC 2 = 42·25 = BC 2.

• We say that, by the CONVERSE of Pythagoras’ Theorem, the triangle is proven to be right angled at A.

BC

A

6·5 cm

5·2 cm3·9 cm

18 cm

19·5 cm

7·5 cmP

Q

R

6·6 cm

11·1 cm

8·8 cm

U

W

V

3. Decide which of these are or are not right angled triangles :-

(a) (b)

4. A groundsman wishes to make sure the footballpitch is “rectangular”.

To check, he measures the diagonal length.Is the pitch rectangular ?

5. Has this flagpole beenerected correctly, so that it is vertical ?

84 mm

91 mm

35 mm

9·6 m

20·4 m

18·7 m

105 m

84 m

63 m

13·5 m 10·8 m

8·1 m

Exercise 4·2

N5 - Chapter 4 this is page 29 Pythagoras

1. (a) Calculate the length of the face diagonal EG of this cuboid.

(b) Now calculate the length of the space diagonal EX.

2. Calculate the length of the face diagonal ACof this water tank, and then calculate the length of the space diagonal AH.

3. Make a sketch of this shoe box and show, usingtwo steps, how to calculate the length of its spacediagonal. (You may want to letter the vertices).

4. Calculate the length of the space diagonal RY of this cube.

Pythagoras’ Theorem applied to 3-Dimensional Problems

Pythagoras’ Theorem only works on a right angled triangle, but right angled triangles appear in 3 Dimensional situations as well.

Exercise 4·3

Example :- Calculate the length of the space diagonal of this cuboid.

Note :- A face diagonal is one joining the opposite vertices of any rectangular face of the cuboid (e.g. BD).A space diagonal is one joining one vertex of the cuboid to the furthest away vertex (e.g. BP).

Solution :- The answer is found by following 2 steps :-

Step 1 :- Space diagonal BP is the longest side in R.A.T. BPD.To find BP, we must first find BD in R.A.T. BAD.

Step 2 :- Now find BP, the longest side in R.A.T. BPD.

A

P S

RQ

D C

B8 cm6 cm

5 cm

BD2 = BA2 + AD2

BD2 = 82 + 62 = 64 + 36 = 100BD = √100 = 10 cm

BP2 = BD2 + DP2

BP2 = 102 + 52 = 100 + 25 = 125BD = √125 = 11·2 cm

E

XU

WV

H G

F5 cm

3 cm 12 cm

80 cmA

E

GF

H

C

B

75 cm

60 cm

30 cm

15 cm

18 cm

10 cm

R

X

S

U T

W

V Y

N5 - Chapter 4 this is page 30 Pythagoras

5 Shown is a square based pyramid ABCDV.

Height MV = 9 cm.

(a) Calculate the length of the diagonal AC.

(b) Write down the length of AM.

(c) Calculate the length of the sloping edge AV.

6. This Popcorn box is in the shape of a pyramid.

The square top has sides 10 cm, and the sloping edge is 13 cm long.

(a) Calculate the length of the diagonal of the open top.

(b) Calculate the height of the box.

(c) Calculate the volume of the box, in cm3.

7. The side face of this wedge is in the shape of a right angled triangle.

(a) Calculate the length of the face diagonal of the base of the wedge.

(b) Now calculate the length of the diagonal of the sloping edge. (The red line).

AM

V

D

C

B

6 cm

13 cm

10 cm 10 cm

12 cm8 cm

6 cm

8. A cone has a base diameter of 10 cm and its sloping edges are 13 cm long.

(a) Calculate the heightof the cone.

(b) Now calculate thevolume of the cone.

9. This empty tin of McTivies biscuits measures24 cm by 24 cm by 16 cm high.

Would this wooden rod, 40 cm long, be able tofit in the box and still allow the tin to be fully closed with its lid on ?

10. Just as you can plot a point in 2-dimensions using 2 coordinates A(4, 3), you can plot points in 3-dimensions like A(4, 3, 1). (See below).

In the above figure, A is given by A(4, 3, 1).(4 right, 3 back and 1 up). AB is parallel to the x-axis.

The cuboid measures 8 by 3 by 5 boxes.

(a) Write down the coordinates of the other 7 points making up the cuboid.

(b) Calculate the length of the diagonal AC.

(c) Calculate the length of space diagonalAS.

(d) Harder. Calculate the length of the line OS.

10 cm

13 cm

A B

CD

Ox

z

yQR

SP

4

1

38

5

3

Volumeprism = Areabase x height13

N5 - Chapter 4 this is page 31 Pythagoras

Remember Remember..... ?Remember Remember..... ? Topic in aNutshell

1. Calculate the lengths of the sides marked x (a) (b)and y, correct to 3 significant figures.

2. Plot the points A(–5, –7) and B(6, 1) on a coordinate diagram and calculate the length of the line AB.

3. Use the Converse of Pythagoras’ Theorem to decide which, if any, of these triangles is/are right angled :-

(a) (b) (c)

4. This figure looks like a kite, but is it really one ?Prove whether it is or is not a kite.

5. A girl made a simple model house out of a cardboard box.

(a) Calculate the length of the red dotted line.

(b) Calculate the length of the space diagonal of the box.

6. Shown is a right angled triangular prism. (a wedge).Use Pythagoras’ Theorem twice to calculate thelength of the sloping dotted blue line.

7. Shown is a square based pyramid and a cone.By calculating the height of both, decide which has the greater volume and by how much.

7·1 cm

x cm7·1 cm

11·6 m

13·8 my m

9·3 cm

15·5 cm

12·4 cm

X

Y

Z

24 m

39 m

32 mP

Q

R

96 mm

104 mm

40 mm

A

C

B

65 mm

52 mm

39 mm

95 mm 39 mm

16 cm

20 mm30 cm

17 cm8 cm

11 cm

8 cm

10 cm

12 cm

15 cm

8 cm

Exercise 4·11 . a 11·3 cm b 11·7 cm c 6·19 m d 5·66 mm2 . a 24 cm b 240 cm2 3 . 1512 cm2 4 . 40 cm5 . x = 86 . a 48 cm b 960 cm2 7 . 11·4 boxes8 . 13·9 boxes9 . a 6 cm b 60 cm3 10. x, being smaller side, should end up less than 1011. 228 mm12. 31 cm

Exercise 4·21 . PQ2 + QR2 = 324 + 56·25 = 380·5 = PR2

By the Converse of Pythagoras’ Theorem, it IS a RAT.2 . 6·62 + 8·82 = 43·56 + 77·44 = 121 ≠ 123·21 = 11·12

Therefore, it IS NOT a RAT3 . a yes b no4 . 842 + 632 = 7056 + 3939 = 11025 = 1052

By the Converse of Pythagoras’ Theorem, it IS a RAT5 . 8·12 + 10·82 = 65·61 + 116·64 = 182·25 = 13·52

By the Converse of Pythagoras’ Theorem, it IS a RAT and the flagpole could be vertical.

Exercise 4·31 . a 13 cm b 13·3 cm2 . AC = 100 cm => AH = 125 cm3 . 38·1 cm4 . 17·3 cm5 . a 8·49 cm b 4·24 cm c 9·95 cm6 . a 14·1 cm b 10·9 cm c 364 cm3 7 . a 14·4 cm b 15·6 cm 8 . a 12 cm b 314 cm3 9 . Space diagonal = 37·5 cm. 40 cm rod is too long to fit in10. a B(12, 3, 1), C12, 6, 1), D(4, 6, 1), P(4, 6, 6)

Q(4, 3, 6), R(12, 3, 6), S(12, 6, 6)b 8·54 units c 9·90 units d 14·7

Remember, Remember1 . a 10·0 cm b 7·48 m2 . 13·6 boxes3 . a 962 + 402 = 9216 + 1600 = 10816 =1042

By the Converse of Pythagoras’ Theorem, it is a RATb 242 + 322 = 576 + 1024 = 1600 ≠ 1521 = 392

Therefore, it IS NOT a RATc 9·32 + 12·42 = 86·49 + 153·76 = 240·25 = 15·52

By the Converse of Pythagoras’ Theorem, it is a RAT4 . 392 + 522 = 1521 + 2704 = 4225 = 652

This means the angle between the diagonals = 90°Because the 2 smaller parts of one diagonal are equal,this means the longer diagonal is a line of symmetry,which means it IS a kite.

5 . a 34 cm b 39·4 cm6 . 21·8 cm7 . Height of pyramid = 13·9 cm, Height of cone = 8 cm

Volume of pyramid = 296·4 cm3, Volume of cone = 301·4 cm3 Cone has bigger volume by 5 cm3 approximately

N5 - Chapter 4 this is page 32 Pythagoras

Chapter 4 - Vectors

Answers AnswersChapter 4Chapter 4