Pythagoras' Theorem
Further Work with Pythagoras' Theorem

Worked Examples

1

Find the length of the side marked x in the diagram.

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First consider triangle ABC. The unknown length of the hypotenuse has been marked y.

y2 = (4 cm)2 + (4 cm)2
y2 = 16 cm2 + 16 cm2
y2 = 32 cm2

Triangle ACD can now be considered, using the value for y2. From the triangle, x2 = y2 + 22 , and using y2 = 32 cm2 gives

x2 = 32 cm2 + 4 cm2
x2 = 36 cm2
x = 36 cm2
x = 6 cm

Note

When finding the side x, it is not necessary to find , but to simply use y2 = 32 cm2 .
2

Find the value of x as shown on the diagram.

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Using Pythagoras' Theorem gives

132 = (2x)2 + (3x)2
169 = 4x2 + 9x2 (since (2x)2 = 22 x2 = 4x2 )
169 = 13x2
13 = x2
x =
= 3.61 m (to 2 decimal places)

Exercises

On this page give your answers rounded to 2 decimal places when necessary, unless otherwise instructed.

Find the length of the side marked x in each diagram.

(a)
x =
(b)
x =
(c)
x =
(d)
x =
(e)
x =
(f)
x =

Find the length of the side marked x in the following situations.

(a)
x =
(b)
x =
(c)
x =
(d)
x =

Which of the following triangles are right angled triangles?

A ladder of length 4 metres leans against a vertical wall. The foot of the ladder is 2 metres from the wall. A plank that has a length of 5 metres rests on the ladder, so that one end is halfway up the ladder.

(a)
How high is the top of the ladder? m
(b)
How high is the top of the plank? m
(c)

How far is the bottom of the plank from the wall?

m

The diagram shows how a cable is attached to the mast of a sailing dingy. A bar pushes the cable out away from the mast. Find the total length of the cable.

cm

A helicopter flies in a straight line until it reaches a point 20 km east and 15 km north of its starting point. It then turns through 90° and travels a further 10 km.

(a)

How far is the helicopter from its starting point?

km
(b)

If the helicopter turned 90° the other way, how far would it end up from its starting point?

km

A thin steel tower is supported on one side by two cables. Find the height of the tower and the length of the longer cable.

Height of the tower: m

Length of the longer cable: m

An isosceles triangle has two sides of length 8 cm and one of length 4 cm. Find the height of the triangle and its area.

Height = cm

Area = cm2

Find the area of each the equilateral triangles that have sides of lengths

(a)
8 cm cm2
(b)
20 cm cm2
(c)
2 cm cm2