A right triangle or right-angled triangle is a triangle in which one of the angles is $90^\circ$90°, in other words a right angle. In diagrams, the $90^\circ$90° angle is indicated by a small square. The longest side of a right triangle is always the one which is opposite the right angle, and it is called the hypotenuse.

Pythagoras' theorem

Pythagoras' theorem states that for a right-angled triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two smaller sides.

Referring to the diagram below, the theorem can be written algebraically:

$a^2+b^2=c^2$a2+b2=c2

where $c$c represents the length of the hypotenuse and $a$a, $b$b are the lengths of the two shorter sides.

How do we know?

We can't construct and measure every possible right triangle to check that the theorem holds, so how do we know it is always true?

Pythagoras was an ancient Greek philosopher who lived about $2500$2500 years ago. Although the theorem is named after Pythagoras, it was actually known and used by ancient Babylonians, Hindus and Chinese centuries before his time! This was quite a long time before any of the modern algebraic mathematics was developed, and so it is likely that these ancient mathematicians would have come to understand the theorem by simple geometric observations such as those in the interactive demonstration below.

Created with Geogebra

This demonstration shows that if $a$a, $b$b and $c$c are the sides of a right triangle, and we break up a square with area $a^2$a2 into $4$4 pieces and put them together with a square of area $b^2$b2, then we can make a square with area $c^2$c2, that is: $a^2+b^2=c^2$a2+b2=c2 !

Practice questions

Question 1

Which side of the triangle in the diagram is the hypotenuse?

$AB$AB
A
$CA$CA
B
$BC$BC
C

Question 2

Find the length of the hypotenuse, $c$c in this triangle.

question 3

Calculate the value of $b$b in the triangle below.

Give your answer correct to two decimal places.

A right-angled triangle is presented with the right angle at the top left. The side adjacent the right angle is labeled with the variable "b." The triangle's height, on the left side, is marked with the number $10$ , and the hypotenuse, slanting down from the left to the right, is labeled with the number $24$ .

question 4

Use Pythagoras' theorem to determine whether this is a right-angled triangle.

A triangle with labeled side lengths. The triangle has sides with different lengths and with no implicit indication of angle measurements. The sides are measured with lengths of $10$10 units, $18$18 units and $21$21 units.

Let $a$a and $b$b represent the two shorter side lengths. First find the value of $a^2+b^2$a2+b2.
Let $c$c represent the length of the longest side. Find the value of $c^2$c2.
Is the triangle a right-angled triangle?
Yes
A
No
B

Question 5

A square prism has sides of length $11$11cm, $11$11cm and $15$15cm as shown.

If the diagonal $HF$HF has a length of $z$z cm, calculate the exact length of $z$z, leaving your answer in surd form.
Now, we want to find $y$y, the length of the diagonal $DF$DF.

Calculate $y$y to two decimal places.

Outcomes

ACMGM017

review Pythagoras’ Theorem and use it to solve practical problems in two dimensions and for simple applications in three dimensions

5.01 Pythagoras' Theorem