MATHEMATICS DEPARTMENT · 2. The theorem 4 and 5 3. Proof of Pythagoras 5 4. Proving a triangle is a right-angled triangle 6 5. Pythagoras’s Theorem – finding the Hypotenuse 6

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MATHEMATICS DEPARTMENT

Knowledge Book

Year 9

Pythagoras’ Theorem

Name: _______________________

Teacher: _____________________

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Contents

Page Number

1. History and background 3 and 4

2. The theorem 4 and 5

3. Proof of Pythagoras 5

4. Proving a triangle is a right-angled triangle 6

5. Pythagoras’s Theorem – finding the Hypotenuse 6

6. Converse of Pythagoras – finding the shorter side 7

7. Practical applications (word problems) 7 and 8

8. Pythagorean Triples 9

9. Applying Pythagoras in 3 dimensions 9

10. Resources 10

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1. Background

Pythagoras’ Theorem is named after the Greek Mathematician

Pythagoras who lived on the Island of Samos off the coast of Turkey

around 2500 years ago. Although Pythagoras is credited with

discovering the theorem there is evidence to suggest that it might

have been known to the Babylonian, Egyptian and Chinese civilizations

at the same time or earlier. However, it was Pythagoras who recorded

the theorem in the form that we understand today.

…. some information about Pythagoras

Pythagoras of Samos was a famous Greek mathematician and philosopher

who lived around 500 years before the Christian era. He is best known for

his proof of the Pythagorean theorem, which uses a property of a right-

angled triangle to find the length of an unknown side.

As well as Pythagoras Theorem, he researched irrational numbers, proved

that the angles of a triangle added up to 180°, is credited with inventing

the Chromatic Scale in Music and stated that the Earth was a sphere at the centre of the Universe.

He founded a group of mathematicians, called the Pythagoreans, who worshipped numbers and lived

like monks. He had an influence on other famous Greek mathematicians such as Plato and Euclid. He

was one of the greatest thinkers of his time.

There is not much information about Pythagoras’ life. It is said that

he had a good childhood. Growing up with two or three brothers,

he was well educated. He did not agree with the government and

their schooling and so moved to Croton to set up his own cult

(little society) of followers under his rule. His followers did not

have any personal possessions, and they were all vegetarians.

Pythagoras taught them all, and they had to obey strict rules.

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Some say he was the first person to use the term philosophy. Since he worked very closely with his

group, the Pythagoreans, it is sometimes hard to tell his works from those of his followers. Religion

was important to the Pythagoreans. They swore their oaths by "1+2+3+4" (which equals 10). They also

believed the soul is immortal and goes through a cycle of rebirths until it can become pure. They

believed that these souls were in both animal and plant so was math life.

2. The Theorem

When a triangle has a right angle (90°) ...

... and squares are made on each of the three sides....

... then the biggest square has the exact same area as the other two squares put together!

The hypotenuse is the side opposite the right angle

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3. The Proof

This algebraic proof is one of many. You will not be asked to reproduce this proof.

Let’s look at some more examples of Pythagoras’ Theorem might be used

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4. Proving that a triangle contains a right angle Example prove that triangle ABC is a right-angled triangle where length AC = 3.4 m, AB = 6.2 m and BC = 7.07 m For a right-angled triangle with BC as the hypotenuse: AC2 + AB2 = BC2

AB2 =38.44 m2 AC2 = 11.56 BC2 = 49.99 = 50 (2 s.f) 38.44 + 11.56 = 50 Therefore, ABC is right angled triangle (BAC is a right angle

5. Finding the Hypotenuse Example Triangle PQR has lengths PQ = 5 m, QR = 12 m. Angle PQR is a right angle. Find the length of the hypotenuse PR

Pythagoras: PR2 = PQ2 + QR2 So: PR = �𝑃𝑃𝑃𝑃2 + 𝑃𝑃𝑄𝑄2 PR = √52 + 122 PR = √25 + 144 PR = √169 PR = 13 m

AB = 6.20 m

AC = 3.40

B A

C

BC = 7.07

QR = 12 m

PQ = 5 m

R Q

P

PR =?

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6. The converse of the Theorem

The length of a shorter side of a right-angled triangle can be found using the length of the other two sides. Instead of adding the lengths of the two shorter sides, we subtract the square of the known shorter side from the square on the hypotenuse.

Example Triangle WXY has lengths WX = 3 m, the hypotenuse WY = 5 m. Angle WXY is a right angle. Find the length of the side Pythagoras: WY2 = WX2 + XY2 (XY is the unknown side)

So we use the converse: XY2 = WY2 – WX2

So: XY = √𝑊𝑊𝑊𝑊2 − XY2 XY = √52 − 32 XY = √25 − 9 XY = √16 XY = 4 m

7. Practical applications of Pythagoras

Pythagoras’ Theorem is one of the most used equations in Science and Engineering.

Some quotes from tradesmen and scientists

“I use it every day, in some form or other. I am a fabricator and welder for a living, and I have to be sure things I make are square and meet the blueprint specifications. I also use it to lay out the corners for any square or rectangular thing, like a patio, a deck, a square or 4-sided piece of anything. Commonly, when I measure the cross angles of something, I can check it’s dimensions with the math to be sure it meets design.

I have heard this referred to as the “3 4 5” method, but it is still the Pythagorean Theorem to me”

XY = ? m

WX = 3 m

Y X

W

WY = 5 m

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“Engineering and science are part of real life. It (sic) is a fundamental theorem used throughout engineering and science.

The Pythagorean theorem is used in most of mathematics. So, for example, it’s used in probability and statistics, and those are used in social scientists, business, as well as engineering and other sciences.

Most of modern society relies somehow or other on the Pythagorean theorem.”

“Suppose you are building a garage that is to be a rectangle 30 ft x 40 ft. It’s easy to make sure the sides are the right lengths, but how do you know the angles are 90 degrees? A non-square parallelogram could also have the same lengths of sides. Builders measure the diagonal and make sure it is 50 ft. By the Pythagorean Theorem, any triangle with sides proportional to a 3x4x5 triangle is a right triangle: 3^2 + 4^2 = 9 + 16 = 25 = 5^2. The principle can be used on a rectangle of any size, of course. Carpenters and builders use this idea all the time”

Example

A rectangular swimming pool is 21 metres wide and 50 metres long. Calculate the length of the diagonal to 1 decimal place.

Always draw a diagram

Pythagoras: PR2 = 502 + 212 (can be written either way round) So: diagonal = √502 + 212 diagonal = √2500 + 441 diagonal = √2941 𝑚𝑚 diagonal = 54.4 m (1d.p)

21 m

50 m

? m

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8. Pythagorean triples

Pythagorean triples are groups of 3 whole number integers where the squares of two of the integers add up to the square of the third. Some examples are: There is an algebraic method which enables Pythagorean triples to be calculated 9. 3-dimensional Pythagoras

In the example above AD could be found using a single step:

𝐴𝐴𝐴𝐴 = �𝐴𝐴𝐴𝐴2 + 𝐴𝐴𝐵𝐵2 + 𝐵𝐵𝐴𝐴2

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PYTHAGORAS WORKSHEET

1) Find the hypotenuse of the following triangles.

a 12cm

8cm c 9cm

15cm 6cm b

b d

8cm 12cm

16cm

2) Using Pythagoras, find the lengths of the sides labelled with letters.

15cm 5cm

b c 12cm

a 17cm 4cm 20cm

12cm d 13cm

3) Find the missing lengths of the triangles below. If necessary, round answers to 1 decimal place.

8cm 6cm

3.4cm a

9cm b 17cm 10cm c

1.9cm e

d 15cm 9cm 20cm f

2.4cm

16cm

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4) Find the perimeter of the following triangles.

5) The triangle DEF is shown below. Find the length of EF to 1 decimal place.

D

7.9cm 8.9cm

E F

6) To wash a window that is 8 metres off the ground, Ben leans a 10 metre ladder against the side of the building. To reach the window, how far from the building should Ben place the base of the ladder?

7) A rectangular swimming pool is 21 metres wide and 50 metres long. Calculate the length of the diagonal to 1 decimal place.

8) Miss Barker is teaching a 5th grade class. She is standing 12 feet in front of Jim. Francisco is sitting 5 feet to Jim’s right. How far apart are Miss Barker and Francisco?

9) A triangle has sides with lengths of 10 metres, 16 metres and 20 metres. Is it a right angled triangle? Explain your reasoning.

a

2 cm

10 cm b

c

6 cm

10 cm

12 cm 13 cm

d

8 cm 15 cm

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10) a) One side of a right angled triangle is 10cm. The other two are both of length x. Calculate x to 2 decimal places.

b) Find the perimeter of the triangle in part a)

11) Find the length of the diagonal of a square of side 4cm to 2 decimal places.

12) The diagram below shows a shaded parallelogram drawn inside a rectangle. Using Pythagoras, find the hypotenuse of triangle A and the hypotenuse of triangle B to 1 decimal place.

3cm

5cm

3cm

10cm

13) Here is a trapezium, use Pythagoras’ Theorem to find the value of k to 1 decimal place.

K cm

20cm 22cm

30cm

A

B

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14) The following triangle is NOT a right-angled triangle and so you cannot apply Pythagoras’ theorem directly. Find the length of x to 2 d.p.

x x 35cm

100cm

Answers:

1) a = 17cm b = 10cm c = 15cm d = 20cm

2) a =8cm b =3cm c =16cm d =5cm

3) a =9.6cm b =15cm c =8cm d =3.1cm e =12cm f =12cm

4) a =48cm b =24cm c =30cm d =40cm

5) EF = 16.8cm

6) 6 metres

7) 54.2 metres

8) 13 ft

9) No. Using Pythagoras, 𝑎𝑎2 + 𝑏𝑏2 ≠ 𝑐𝑐2 (a squared + b squared does not equal hypotenuse squared)

10) a) x = 7.07cm b) perimeter = 24.14 cm

11) 5.65 cm

12) A = 5.8cm B = 5.8cm

13) k = 20.8cm

14) x= 61.03 cm