Mathematics Form 3(Phythagoras Theorem)

8/3/2019 Mathematics Form 3(Phythagoras Theorem)

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MATHEMATICS FORM 3PHYTAGORAS THEOREMYears ago, a man named Pythagoras found anamazing fact about triangles:

If the triangle had a right angle (90) ...

... and you made a square on each of the three

sides, then ...

... the biggest square had the exact same area as the other two

squares put together!

It is called "Pythagoras' Theorem" and can be written in one

short equation:

a

2

+ b

2

= c

2

Note:

c is the longest side of the triangle

a and b are the other two sides

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:


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In a right angled triangle:

the square of the hypotenuse is equal to

the sum of the squares of the other two sides.

Sure ... ?

Let's see if it really works using an example.

Example: A "3,4,5" triangle has a right angle in it.

Let's check if the areas are the same:

32 + 42 = 52

Calculating this becomes:9 + 16 = 25

It works ... like Magic!

Why Is This Useful?

If we know the lengths oftwo sides of a right angled triangle, we can find the length of

the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

a2 + b2 = c2

Now you can use algebra to find any missing value, as in the following examples:

Example: Solve this triangle.
http://www.mathsisfun.com/geometry/triangle-3-4-5.htmlhttp://www.mathsisfun.com/algebra/index.htmlhttp://www.mathsisfun.com/geometry/triangle-3-4-5.htmlhttp://www.mathsisfun.com/algebra/index.html


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a2 + b2 = c2

52 + 122 = c2

25 + 144 = c2

169 = c2

c2 = 169c = 169c = 13

You can also read about Squares and Square Roots to find out why 169 = 13

Example: Solve this triangle.

a2 + b2 = c2

92 + b2 = 152

81 + b2 = 225Take 81 from both sides:

b2 = 144b = 144b = 12

Example: What is the diagonal distance across a square of size 1?

a2 + b2 = c2

12 + 12 = c2

1 + 1 = c2

2 = c2

c2 = 2c = 2 = 1.4142...

It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2,then the triangle is right angled.

Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ?

a2 + b2 = 102 + 242 = 100 + 576 = 676 c2 = 262 = 676

They are equal, so ...

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?Does 82 + 152 = 162 ?
http://www.mathsisfun.com/square-root.htmlhttp://www.mathsisfun.com/square-root.html


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82 + 152 = 64 + 225 = 289,

but 162 = 256

So, NO, it does not have a Right AngleExample: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ?Does (3)2 + (5)2 = (8)2 ?Does 3 + 5 = 8 ?

Yes, it does!So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

Draw a right angled triangle on the paper, leaving

plenty of space.

Draw a square along the hypotenuse (the longest

side)

Draw the same sized square on the other side of

the hypotenuse

Draw lines as shown on the animation, like this:

Cut out the shapes

Arrange them so that you can prove that the big

square has the same area as the two squares on

the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.


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Watch the animation, and pay attention when the triangles start sliding

around.

You may want to watch the animation a few times to understand what

is happening.

The purple triangle is the important one.

EXAMPLE QUESTIONS ANDWAYS TO OVERCOME IT


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Solve this triangle

A

c = 5

B

c = 25

C

c = 527

D

c = 31a2 + b2 = c2

72 + 242 = c2

49 + 576 = c2

c2 = 625

c = 625

c = 25

or7x24=168/168=25


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Solve this triangle(It is not drawn to scale)

A

a = 5B

a = 35

C

a = 135

D

a = 377

a

2

+ b

2

= c

2

a2 + 112 = 162

a2 + 121 = 256

Take 121 from both sides: a2 = 256 - 121 = 135

a = 135

What is the length of the diagonal of a rectangle of length 3 and width 2?


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A

5

B

13C

5

D

6

The rectangle can be divided up into two right-angled triangles, as shown in thediagram.

We can find the length of the diagonal, d, by using Pythagoras' theorem in one triangle:d2 = 32 + 22 = 9 + 4 = 13

d = 13

Which one of the following triangles is NOT a right triangle?

AWhich one of the following triangles is NOT a right triangle?


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B

C

D

Use the converse of Pythagoras' Theorem: When the threesides of a triangle make a2 + b2 = c2, then the triangle isright angled.

In A, 62 + 82 = 36 + 64 = 100 = 102, so the triangle is a

right triangleIn B, (3)2 + (8)2 = 3 + 8 = 11 = (11)2, so thetriangle is a right triangleIn C, 32 + 52 = 9 + 25 = 34 62, so the triangle is not aright triangleIn D, 32 + (10)2 = 9 + 10 = 19 = (19)2, so thetriangle is a right triangle


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What is the length of the side x?

A

x = 5

B

x = 34C

x = 211

D

x = 261In this question there are two right triangles. Use Pythagoras' Theorem in each of themin turn:

c2 = 62 + 82 = 36 + 64 = 100So c = 100 = 10

Now we know the value of c, mark it in on the second triangle and use Pythagoras'Theorem again:


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122 = 102 + x2

144 = 100 + x2

x2 = 144 - 100 = 44

x = 44 = 211

Which one of the following triangles is a right triangle?

A

A

B

B

C

C


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D

DUse the converse of Pythagoras' Theorem: When the three sides of a triangle make a2 +

b2 = c2, then the triangle is right angled.

In A, (7)2 + (10)2 = 7 + 10 = 17 42, so the triangle is not a right triangleIn B, (10)2 + 32 = 10 + 9 = 19 = (19)2, so the triangle is a right triangleIn C, 42 + 112 = 16 + 121 = 137 122, so the triangle is not a right triangleIn D, 62 + (12)2 = 36 + 12 = 48 72, so the triangle is not a right triangle

The diagram shows a kite ABCD. The diagonals cut at right angles and intersect at O.What is the length of the diagonal AC?

A

16

B

19

C

389

D

21


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Use Pythagoras' Theorem twice:1. In triangle AOD

102 = x2 + 82

100 = x2 + 64

x2 = 100 - 64 = 36

x = 36 = 6

1. In triangle COD

172 = y2 + 82

289 = y2 + 64

y2 = 289 - 64 = 225

y = 225 = 15

Therefore the length of AC = x + y = 6 + 15 = 21


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Town B is 8 miles north and 17 miles east of town A. How far are the two towns apart?

A

15 miles

B

18.5 miles

C

18.8 miles

D

25 milesComplete the right-angled triangle ABC showing that B is 8 miles north and 17 mileseast of A:

We are asked to find the distance from A to B = c miles.

By Pythagoras:c2 = a2 + b2

c2 = 82 + 172 = 64 + 289 = 353

c = 353 = 18.8 correct to one decimal place.

The two towns are 18.8 miles apart.


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A 3m ladder stands on horizontal ground and reaches 2.8 m up a vertical wall. How faris the foot of the ladder from the base of the wall?

A

0.2 m

B

1.08 m

C

1.47 m

D

4.10 mComplete a right-angled triangle ABC showing that the ladder is 3 m long and the

distance up the wall is 2.8 m:

We are asked to find the distance from the foot of the ladder to the base of the wall = "a"m.

By Pythagoras:c2 = a2 + b2

32 = a2 + 2.82


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9 = a2 + 7.84

a2 = 9 - 7.84 = 1.16

a = 1.16 = 1.08 correct to 2 decimal places

The foot of the ladder is 1.08 m from the base of the wall

A rectangular field is 125 yards long and the length of one diagonal of the field is 150yards. What is the width of the field?

A

82.9 yards

B

83.2 yardsC

88.7 yards

D

195.3 yardsThe rectangular field is two right-angled triangles (one of which is triangle ABC):

We are asked to find the width of the field = b yds.

By Pythagoras: c2 = a2 + b2

1502

= 1252

+ b2

22,500 = 15,625 + b2

b2 = 22,500 - 15,625 = 6,875

b = 6,875 = 82.9 correct to 1 decimal place


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The width of the field is 82.9 yards.Question of the day!!!fractions

If 2 is subtracted from the numerator and 1 is added to the denominator,

Find the fraction.

A

B

C

D

Let x be the numerator and y be the denominator of the fraction. The fraction is

--------------(1)

---------------(2)

Solving for x and y using equations (1) and (2),


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5x - 2y = 9-6x +2y = -14-x = -5, x = 5.25 - 2y = 9

-2y = 9 - 25 = -16y = 8.

Therefore, the fraction is

Mathematics Form 3(Phythagoras Theorem)

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