Sec 2 · G3 (Express)

Pythagoras' theorem

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1. Pythagoras' theorem

Find	Formula	Example
the hypotenuse $c$	$c^{2} = a^{2} + b^{2}$	$a = 5$ , $b = 12$ : $c = 13$
a shorter side $a$	$a^{2} = c^{2} - b^{2}$	$c = 10$ , $b = 6$ : $a = 8$

Hypotenuse: the side opposite the right angle

\displaystyle c^{2} = 5^{2} + 12^{2}

e.g.

\displaystyle c = \sqrt{5^{2} + 12^{2}}

\displaystyle c = \sqrt{169}

\displaystyle c = 13

A shorter side: subtract

\displaystyle a^{2} = 9^{2} - 4^{2}

e.g.

\displaystyle a = \sqrt{9^{2} - 4^{2}}

\displaystyle a = \sqrt{65}

\displaystyle a = 8.06\text{ (3 s.f.)}

📖 Worked example

In the diagram, $\angle XYZ = 90^\circ$ . Find $x$ .

Step 1 of 3

Name the hypotenuse: the side opposite the right angle

The hypotenuse is

XZ

, opposite the right angle at

Y

📖 Worked example

In the diagram, $\angle LMN = 90^\circ$ . Find $x$ .

Step 1 of 3

Name the hypotenuse: the side opposite the right angle

The hypotenuse is

LN

, opposite the right angle at

M

📖 Worked example

A ladder $7$ m long leans against an upright wall. Its foot is on level ground, $1.5$ m from the wall. Find the height, $h$ m, that the ladder reaches up the wall, correct to 3 significant figures.

Step 1 of 2

Name the hypotenuse: opposite the right angle between the upright and the ground

The hypotenuse is the ladder, opposite the right angle at the foot of the wall.

Practise this →

2. Is the triangle right-angled?

Sides	Check the longest	Right-angled?
$8, 15, 17$	$8^{2} + 15^{2}$ is $289$ , and $17^{2}$ is $289$	yes
$6, 7, 9$	$6^{2} + 7^{2}$ is $85$ , and $9^{2}$ is $81$	no

📖 Worked example

In triangle $LMN$ , $LM = 48$ cm, $MN = 60$ cm and $LN = 36$ cm. Determine whether triangle $LMN$ is a right-angled triangle.

Step 1 of 4

Square each side

LM^2 = 48^2

= 2304

MN^2 = 60^2

= 3600

LN^2 = 36^2

= 1296

📖 Worked example

In triangle $XYZ$ , $XY = 9$ cm, $YZ = x$ cm and $XZ = y$ cm, where $YZ$ is the longest side. It is given that $(x + y)(x - y) = 81$ . Determine whether triangle $XYZ$ is right-angled. Choose the answer with its reason.

Step 1 of 3

Expand the brackets: (x + y)(x − y) is a difference of two squares

(x + y)(x - y) = x^2 - y^2

, so

x^2 - y^2 = 81

📖 Worked example

In the diagram, $\angle KLM = 90^\circ$ , $KL = 8$ cm and $KM = 10$ cm. $N$ is the point on $KM$ such that $LN$ is perpendicular to $KM$ . Find the shortest distance from $L$ to $KM$ , $h$ cm.

Step 1 of 3

First find LM: KM is the hypotenuse, so SUBTRACT, then square-root

LM^2 = 10^2 - 8^2

= 36

LM = \sqrt{36}

= 6

Practise this →

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1. Pythagoras' theorem

Find	Formula	Example
the hypotenuse $c$	$c^{2} = a^{2} + b^{2}$	$a = 5$ , $b = 12$ : $c = 13$
a shorter side $a$	$a^{2} = c^{2} - b^{2}$	$c = 10$ , $b = 6$ : $a = 8$

Hypotenuse: the side opposite the right angle

\displaystyle c^{2} = 5^{2} + 12^{2}

e.g.

\displaystyle c = \sqrt{5^{2} + 12^{2}}

\displaystyle c = \sqrt{169}

\displaystyle c = 13

A shorter side: subtract

\displaystyle a^{2} = 9^{2} - 4^{2}

e.g.

\displaystyle a = \sqrt{9^{2} - 4^{2}}

\displaystyle a = \sqrt{65}

\displaystyle a = 8.06\text{ (3 s.f.)}

📖 Worked example

In the diagram, $\angle XYZ = 90^\circ$ . Find $x$ .

Step 1 of 3

Name the hypotenuse: the side opposite the right angle

The hypotenuse is

XZ

, opposite the right angle at

Y

📖 Worked example

In the diagram, $\angle LMN = 90^\circ$ . Find $x$ .

Step 1 of 3

Name the hypotenuse: the side opposite the right angle

The hypotenuse is

LN

, opposite the right angle at

M

📖 Worked example

A ladder $7$ m long leans against an upright wall. Its foot is on level ground, $1.5$ m from the wall. Find the height, $h$ m, that the ladder reaches up the wall, correct to 3 significant figures.

Step 1 of 2

Name the hypotenuse: opposite the right angle between the upright and the ground

The hypotenuse is the ladder, opposite the right angle at the foot of the wall.

Practise this →

2. Is the triangle right-angled?

Sides	Check the longest	Right-angled?
$8, 15, 17$	$8^{2} + 15^{2}$ is $289$ , and $17^{2}$ is $289$	yes
$6, 7, 9$	$6^{2} + 7^{2}$ is $85$ , and $9^{2}$ is $81$	no

📖 Worked example

In triangle $LMN$ , $LM = 48$ cm, $MN = 60$ cm and $LN = 36$ cm. Determine whether triangle $LMN$ is a right-angled triangle.

Step 1 of 4

Square each side

LM^2 = 48^2

= 2304

MN^2 = 60^2

= 3600

LN^2 = 36^2

= 1296

📖 Worked example

Step 1 of 3

Expand the brackets: (x + y)(x − y) is a difference of two squares

(x + y)(x - y) = x^2 - y^2

, so

x^2 - y^2 = 81

📖 Worked example

In the diagram, $\angle KLM = 90^\circ$ , $KL = 8$ cm and $KM = 10$ cm. $N$ is the point on $KM$ such that $LN$ is perpendicular to $KM$ . Find the shortest distance from $L$ to $KM$ , $h$ cm.

Step 1 of 3

First find LM: KM is the hypotenuse, so SUBTRACT, then square-root

LM^2 = 10^2 - 8^2

= 36

LM = \sqrt{36}

= 6

Practise this →