G 22

Pythagoras’ Theorem

Subject Content References: G2.1, G2.1h

GCSE Maths Geometry & Measures

Pythagoras’ theorem states:

For any 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

Let’s take a look at a right-angled triangle with sides 3, 4 & 5 cm:

The hypotenuse is 5 cm long, so the area of the square on the

hypotenuse is 25 cm2

This side is 3 cm long, so the area of the square on this

side is 9 cm2

. . and this side is 4 cm long,so the area of the square on this

side is 16 cm2

9

16

25 As we can see,

9 + 16 = 25

In general, for any right-angledtriangle of sides a, b & c, where c

is the hypotenuse

a2 +b2 = c2

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Example

Clearly, this works for a ‘3, 4, 5’ triangle, but does it work for all right angled triangles?

Let’s have a look . . here’s a right-angled triangle:

cbaI measured a, b and c and found:

a = 5.8 cm, b = 2.9 cm and c = 6.5 cm

Theoretically (using Pythagoras’ theorem), 5.82 + 2.92 should be equal to 6.52

5.82 + 2.92 = 33.64 + 8.41 = 42.05

. . which is pretty close

In order to verify Pythagoras’ theorem for ourselves, draw two or three right angled triangles on squared paper and measure their sides as accurately as possible. To check, use the formula a2 + b2 = c2 for each triangle drawn

Since for any right-angled triangle, a2 + b2 = c2 (Pythagoras)

we can say a2 = c2 - b2 (subtracting b2 from both sides), and

This is really important, because it means that if we know the lengths of any two sides of a right-angled triangle, we can calculate the length of the third side . .

b2 = c2 - a2 (subtracting a2 from both sides)

The actual value of 6.52 = 42.25

. . and my measuring may not have been accurate enough . .

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Example

Exercise 1

Example

Examples

1) A right-angled triangle ABC has sides a = 8cm and b = 4cm. Calculate the length of side c to 1 decimal place:

ABcabCIf a2 + b2 = c2 (Pythagoras)

then 82 + 42 = c2

so 64 + 16 = c2 c2 = 80

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⇒

€

⇒ c = √80

c = 8.9cm (1 dp)

2) A right-angled triangle ABC has sides a = 8cm and c = 11cm. Calculate the length of side b to 1 decimal place:

ABcabCIf a2 + b2 = c2 (Pythagoras)

then b2 = c2 - a2 (subtracting a2 from both sides to make b the subject of the equation)

so b2 = 112 - 82

€

⇒ b2 = 121 - 64

€

⇒ b = √57

b = 7.5cm (1 dp)

answer

answer

Exercise 2

Using a right-angled triangle with sides a, b and c, where c is the hypotenuse, calculate (to 1 dp) the following:

3) c, when a = 7km, b = 6km

1) a, when b = 7cm, c = 10cm

2) b, when c = 9m , a = 6m

4) b, when c = 12m , a = 8m

5) c, when a = 3km, b = 2km

6) a, when b = 1.3cm, c = 5.1cm

7) c, when a = 3.8km, b = 4.5km

8) a, when b = 8.9cm, c = 13.4cm

Example: Examination question

A ladder of length 6m is leant up against the top of a vertical wall. It’s base is 2.7m away from the wall. Calculate the height of the wall to the nearest cm:

Step 1:

6m

2.7m

Step 2:

hIn any right-angled triangle with sides a, b and c, where c is the hypotenuse,

a2 + b2 = c2 (Pythagoras)

h2 + 2.72 = 62

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⇒ h2 = 62 - 2.72

h2 = 36 - 7.29 = 28.71

h = √28.71 = 5.36m answer

Draw the diagram

We need to show that we recognise this as a problem involving Pythagoras’ theorem, so state it:

From the diagram, therefore,

. . subtracting 2.72 from both sides

Exercise 3

1) Point A is due north of point B at a distance of 12 km. Point C is due east of point B at a distance of 7 km. Calculate the distance between point C and point A to the nearest metre.

2) The hypotenuse of a right-angled triangle measures 22.3 cm. Its height measures 15.4 cm. Calculate the length of the base of the triangle to the nearest millimetre.

3) Look at the following grid (divided into equal squares). If each square is of side 1 metre, calculate the distance a) between A & B and b) between C & D to the nearest cm:

A

B

C

D

a)

b)

Pythagoras’ theorem can also be used to solve 3-D problems that usually involve more than one triangle . .

Below is a cuboid of width 5.8 cm, length 8.6 cm and height 3.5 cm.

Example

Calculate the length of the line BH to the nearest mm:

AB

C D

EF

G H5.8 cm

8.6 cm

3.5 cm

Step 1: In order to calculate the length of line BH, we need to know the length of line CH . .

. . because the right-angled triangle BCH willenable us to find BH using Pythagoras’ theorem

Step 2: Identify the right-angled triangle CGH . .

. . because the right-angled triangle CGH willenable us to find CH using Pythagoras’ theorem

Step 3: Using Pythagoras’ theorem . .

(CH)2 = 8.62 + 5.82

= 73.96 + 33.64

CH = √107.6 = 10.373 cm

Step 4: Using Pythagoras’ theorem . .

(BH)2 = 3.52 + 10.3732

= 12.25 + 107.6

BH = √119.85 = 10.9 cm answer

Exercise 4

1) Below is drawn a cuboid shaped room of height 2.2 m, width 5.6 m and length 9.7 m. Calculate the distance BS to the nearest cm:

A B

CD

P Q

RS

2) This a cuboid shaped factory of height 14.5 m, width 30.6 m and length 72.7 m. Calculate the distance AY to the nearest cm:

A B

CDW X

YZ