phythagoras solving a missing side of a right angled triangle. miss hudson’s maths

Phythagoras

Solving a missing side of a

right angled triangle.

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Objective for the next week and a half

- To solve problems involving the Pythagorean Theorem.

For Right Angled Triangles Only!

Hypotenuse - always oppositethe right angle

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Step one - You must always identify which side is the hypotenuse in your

given triangle.

leg

leg

hypotenusea

b

c

This helps us to identify which sides are where in our formula.

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ba

abHypotenuse

c

Pythagorean Theorem2 2 2a b c

In our formula c always represents the longest side known as the hypotenuse and a and b always represent the shorter sides.

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However each right angled triangle can have different letters to identify the sides.

This doesn’t change the formula that we use!

We must still identify the hypotenuse and then the short sides like normal.

B

S

T

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6

8

x

2 2 2a b c 2 2 26 8 x

236 64 x 2100 x

2100 x10 x

x 10 or 10

Solve for x.

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4

7

y

2 2 2a b c Solve for y.

2 2 27 4 y 249 16 y 265 y

265 y

65 y

y 65 or 65 y 8.1

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To the nearest meter, find the length of the diagonal of a rectangle with a width of 4 meters and a length of 10 meters.

4 m.

10 m.

2 2 2a b c 2 2 24 10 x

216 100 x 2116 x

2116 x

x 116

x

x 10.8Miss Hudson’s Maths

20 kms

A car drives 20 kilometres due east and then 45 kilometres due south. To the nearest kilometre, how far is the car from its starting point?

45 kmsx

2 2 2a b c 2 2 220 45 x

2400 2025 x 22425 x

22425 x

x 2425x 49.24

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Written Questions:

A ladder is leaning against a wall. The base of the ladder is 9m away from the base of the wall and the top of the ladder reaches 16m up the wall. What is the length of the ladder?

Hint: Draw a diagram to help your understanding.Give the unknown length of the ladder a pronumeral.

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Written Questions:

On my way to school I walk around a sporting complex. I travel 2.4kms east and then turn south and walk another 3kms. On the way home I decide to take a shortcut through the sporting complex and walk directly home. What distance will I walk when I travel home?

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Written Questions:

In the local Darch park a massive tree has been damaged by a storm. The

Wanneroo shire decides they need to stabilize the tree by attaching wires to

the top of the tree and the ground. If the tree is 17 metres tall and the shire workers want to connect the wires 7.5 meters from the base of the tree how

much wire will they require?

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FINDING THE SHORTER SIDE!

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6

t15

2 2 2a b c

Solve for t.

2 2 2t 6 15 2t 36 225

36 36 2t 1892t 189t 189t 13.7

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Mixed

Problems” -

missing the

hypotenuse or

short side.

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12 cm

5 cmc

c2 = 52 + 122

c2 = 169

c =√169

c = 13 cm

Eg 1: When calculating the length of the hypotenuse, we ADD the squares of the other 2 sides.

Eg 2: When calculating the length of one of the shorter two sides we SUBTRACT the squares.

6 cm

3 cm

x x 2 = 62 - 32

x 2 = 27

x = √27x = 5.2 cm (1dp)

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ApplicationsA ladder is placed up the side of a building. The ladder is 8 m long, and the foot of the ladder is 3 m out from the wall. Draw a diagram to illustrate this, and calculate how far up the building the ladder reaches.

Let x be the height up the building.

x 2 = 82 - 32

x 2 = 55

x = √55 x = 7.42 (2dp)

The ladder reaches 7.42 m up the side of the building.

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A square has a length

of 11mm.Find the length of the

diagonal of the square.

11mm

11mms

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Find the width of a

rectangle with a

length of 20cm and

a diagonal of 24cm.

20cm

w24cm

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A rectangle has a length

of 14cm and a width of

8cm. Find the length of

the diagonals of this

rectangle.14cm

8cm d

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Exte

nsio

n

Questio

nFind the length of a

square which has a

diagonal of 16cm.

Note: Both of the short

sides will be worth the

same value because it is

a square.

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Exte

nsio

n

Questio

nFind the length of a

square which has a

diagonal of 27cm.

Note: Both of the short

sides will be worth the

same value because it is

a square.

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Extension

Question

Find the slant height of a

triangle if the area is

96m2 and has a base of

16m.

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Extension

Question

Find the slant height of a

triangle if the area is

75m2 and has a base of

15m.

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Pyt

hagore

an T

riple

s

3 4 5

6 8 10

9 12 15

12 16 20

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Pythagorean Triples

3 4 5

6 8 10

9 12 15

12 16 20

5 12 13

10 24 26

15 36 39

7 24 25

14 48 50

21 72 75

915

12

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Informal Proof #1

Inscribe a square within the square.

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a b

a

a

a

b

b

b c

c

c

c

2c

Informal Proof #1

2Yellow region c

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a b

a

a

a

b

b

b c

c

c

c

a

b c

2c

Informal Proof #1

2Yellow region c

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b

a

a

a

b

b

c

c

c

a

b c

2c

Informal Proof #1

2Yellow region c

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b

a

a

a

b

b

c

c

c

a

b c

2c

a

bc

Informal Proof #1

2Yellow region c

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b

a

a

b

c

c

a

b c

2c

a

bc

Informal Proof #1

2Yellow region c

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b

a

a

b

c

c

a

b c

2c

a

bc

b

ac

Informal Proof #1

2Yellow region c

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a

b

c

a

b c

2c

a

bc

b

ac

Informal Proof #1

2Yellow region c

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a

b

c

a

b c

2c

a

bc

b

aca

b

c

Informal Proof #1

2Yellow region c

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a

b c

2c

a

bc

b

aca

b

c

Informal Proof #1

2Yellow region c

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a

b c

a

bc

b

aca

b

c

Informal Proof #1

2Yellow region c

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a

b c

a

bc

b

aca

b

c

Informal Proof #1

a

b c

2Yellow region c

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a

bc

b

aca

b

c

Informal Proof #1

a

b c

a

bc

2Yellow region c

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b

aca

b

c

Informal Proof #1

a

b c

a

bc

a

b

c

2Yellow region c

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b

ac

Informal Proof #1

a

b c

a

bc

a

b

c

b

ac

2Yellow region c

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Informal Proof #1

a

b c

a

bc

a

b

c

b

ac

2b

2a

2Yellow region c

2 2Yellow region a b

2 2 2a b c

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a b

a

a

a

b

b

b c

c

c

c

2c

Informal Proof #2

a + bTotal Purple Yellow Area Area Area- =

2(a b) -1

4 ab2

= 2c

2 2a 2ab b 2ab 2c

2 2 2a b c

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