Unit 15 Section 1 : Pythagoras' Theorem

Pythagoras' Theorem describes the important relationship between the lengths of the sides of a right-angled triangle.

Pythagoras' Theorem

In a right-angled triangle,

a2 + b2 = c2

The longest side, c, in a right-angled triangle is called the hypotenuse.

Example 1

Calculate the length of the side AB of this triangle:

In this triangle,
AB2 = AC2 + BC2
= 52 + 92
= 25 + 81
= 106
AB  = = 10.29563014 cm
= 10.3 cm (to 1 decimal place)

Example 2

Calculate the length of the side XY of this triangle.

In this triangle,
YZ2 = XY2 + XZ2
142= XY2 + 62
196= XY2 + 36
XY2 = 160
XY = = 12.64911064 cm
= 12.6 cm (to 1 decimal place)

Example 3

Determine whether or not this triangle contains a right angle.

If the triangle does contain a right angle, then the longest side, BC, would be the hypotenuse.
So, the triangle will be right-angled if AB2 + AC2 = BC2.

First consider,
AB2 + AC2 = 72 + 142 = 49 + 196
= 245
Now consider,
BC2 = 192
= 361
In this triangle,

AB2 + AC2 ≠ BC2

so it does not contain a right angle.

Exercises

Question 1

Calculate the length of the hypotenuse of each of the triangles shown. Where necessary, give your answers correct to 2 decimal places.

(a)
cm
(b)
cm
(c)
cm
(d)
m
Question 2

Calculate the length of the unmarked side of each of the triangles shown. In each case, give your answer correct to 2 decimal places.

(a)
cm
(b)
m
(c)
m
(d)
mm
Question 3

(a)

Determine AB.

cm
(b)

Determine EF.

m
(c)

Determine GH.

km
(d)

Determine JK.

m
Question 4

Which of the triangles below contain right angles?

(a)
(b)
(c)
(d)
Question 5

Sam walks 100 m north and then 100 m east. How far is she from her starting position? Give your answer to a sensible degree of accuracy.

Her distance from the starting point is .
Question 6

Calculate the perimeter of the trapezium shown. Give your answer to the nearest millimetre.

Perimeter: cm (to the nearest mm)
Question 7

The diagram shows a plan for a wheelchair ramp.

The distance AC is 2 m.

Giving your answer in metres, correct to the nearest cm, calculate the distance AB if:

(a)

BC = 20 cm

AB = m
(b)

BC = 30 cm

AB = m
Question 8

Calculate the perimeter and area of this trapezium:

Perimeter = cm

Area = cm²

Question 9

A rope is 10 m long. One end is tied to the top of a flagpole. The height of the flagpole is 5 m. The rope is pulled tight with the other end on the ground.

How far is the end of the rope from the base of the flagpole? Give your answer to a sensible level of accuracy.

The end of the rope is m from the flagpole.
Question 10

A ladder leans against a vertical wall. The length of the ladder is 5 m. The foot of the ladder is 2 m from the base of the wall.

How high is the top of the ladder above the ground? Give your answer to a sensible level of accuracy.

m high
Question 11

Sarah makes a kite from two isosceles triangles, as shown in the diagram.

Calculate the height, AC, of the kite, giving your answer to the nearest centimetre.

AC = cm
AC = +
= +
= 26.45751311 + 74.16198487
= 100.619498 cm = 101 cm (to the nearest cm)
Question 12

Cape Point is 7.5 km east and 4.8 km north of Arton.

Calculate the direct distance from Arton to Cape Point.

Distance from Arton to Cape Point = km (to 1 d.p.)
AC² = 7.5² + 4.8² = 56.25 + 23.04 = 79.29
AC = = 8.904493248 km = 8.9 km (to 1 d.p.)
Question 13

A cupboard needs to be strengthened by putting a strut on the back of it like this.

(a)

Calculate the length of the diagonal strut.

cm (to the nearest cm)
length of strut = = = 242.0743687 cm ≈ 242 cm
(b)

In a small room the cupboard is in this position.

Calculate if the room is wide enough to turn the cupboard like this and put it in its new position.

We need cm to turn the clipboard, so the room is .

diagonal of rectangle = = = 170 cm

Since 170 cm > 165 cm, i.e. diagonal of rectangle > width of room, the room is not wide enough for the cupboard to be moved to its new position.