Hypotenuse and right-angled triangles | Year 8

01 / LEARN

Understand the idea.

Use the angle, not the appearance

Pythagoras' theorem applies to a right-angled triangle, which has one angle of exactly $90^{\circ}$ . The two sides meeting at that angle are the legs. The opposite side is the hypotenuse and is always longest.

Use a right-angle marker or stated information to identify the angle. A sketch that only looks square does not establish a right angle. Rotating the triangle changes its appearance but never changes which side is opposite that angle.

Name the sides before calculating

In $a^{2} + b^{2} = c^{2}$ , the letters $a$ and $b$ name the legs in either order; $c$ names the hypotenuse. The placement of the longest side in the equation matters more than the letter chosen.

Identify the right angle and the side opposite it before substituting numbers. Express all lengths in the same unit. The squares in the theorem compare areas, so adding the unsquared lengths would describe a different relationship.

a^{2} + b^{2} = c^{2}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Read the angle marker

The legs in the supplied right triangle are 6 cm and 8 cm. Which side is the hypotenuse?

Worked solution

1
Locate the marked right angle.
The sides of length 6 cm and 8 cm meet at that angle, so they are the two legs.
2
Identify the side opposite it.
$c$
The remaining side faces the right angle and is therefore the hypotenuse.
3
Check the identification against its size.
$c > 8 cm$
The hypotenuse of a right triangle is longer than either leg, regardless of the drawing's orientation.

Answer

The side opposite the marked right angle, labelled c.

EXAMPLE 02

Do not trust a sketch

A triangle has side lengths 5 cm, 6 cm and 8 cm. Can you assume it is right-angled because 8 is the longest side?

Worked solution

1
Choose the longest side for comparison.
$8 cm$
If this triangle were right-angled, its longest side would have to be the hypotenuse.
2
Square and add the two shorter lengths.
$5^{2} + 6^{2} = 25 + 36 = 61$
A right triangle requires this sum to equal the longest side's square.
3
Compare with the longest square.
$8^{2} = 64 \neq 61$
The required equality fails, so these lengths do not form a right triangle; being longest is insufficient.

Answer

No. Being longest is necessary for a hypotenuse, but does not prove a right angle.

03 / YOUR TURN

Try it yourself.

Check your understanding

The right angle is between sides p and q. Which side is opposite it?

Show answer & reasoning

The third side, the hypotenuse.

The sides $p$ and $q$ meet at the right angle, so they are the legs. The third side lies opposite that angle and is the hypotenuse.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

In the supplied triangle, identify its two legs and its hypotenuse.
2 marks
A right triangle has legs $a$ and $b$ and hypotenuse $c$ . Write its squared-length relationship.
2 marks
A right angle is opposite side $A B$ in triangle $A B C$ . Which side is the hypotenuse?
2 marks
Could a right triangle have hypotenuse 7 cm and a leg of 9 cm? Explain.
2 marks
Can you apply the theorem to a triangle known only to have an angle of 60 degrees?
2 marks
The same right triangle is turned upside down. Does its hypotenuse change? Explain.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

In the supplied triangle, identify its two legs and its hypotenuse.
2 marks
The labelled legs meet at the square marker.
The side across from the marker is $h$ .
Legs 5 cm and 12 cm; hypotenuse $h$ .
A right triangle has legs $a$ and $b$ and hypotenuse $c$ . Write its squared-length relationship.
2 marks
Square each leg length and add.
The result is the square of the hypotenuse.
$a^{2} + b^{2} = c^{2}$ .
A right angle is opposite side $A B$ in triangle $A B C$ . Which side is the hypotenuse?
2 marks
The hypotenuse is opposite the right angle.
The right angle is therefore at $C$ .
$A B$ .
Could a right triangle have hypotenuse 7 cm and a leg of 9 cm? Explain.
2 marks
The hypotenuse must be the longest side.
A 9 cm leg would be longer than 7 cm.
No.
Can you apply the theorem to a triangle known only to have an angle of 60 degrees?
2 marks
A 60-degree angle does not establish a right angle.
You need a right angle or enough information to test for one.
Not from that information alone.
The same right triangle is turned upside down. Does its hypotenuse change? Explain.
2 marks
Rotation keeps side lengths and angles unchanged.
The same side remains opposite the right angle.
No.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics (2026)

Hypotenuse and right-angled triangles; why the theorem works; finding the hypotenuse; finding a shorter side; exact roots versus rounded answers; checking whether a triangle is right-angled; practical length problems.

Official curriculum source ↗

In this lesson

Understand the idea.

Use the angle, not the appearance

Name the sides before calculating

See the method in action.

Read the angle marker

Locate the marked right angle.

Identify the side opposite it.

Check the identification against its size.

Do not trust a sketch

Choose the longest side for comparison.

Square and add the two shorter lengths.

Compare with the longest square.

Try it yourself.

Check your understanding

Keep practising