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 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.
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.
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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 !
Which side of the triangle in the diagram is the hypotenuse?
$AB$AB
$CA$CA
$BC$BC
Find the length of the hypotenuse, $c$c in this triangle.
Calculate the value of $b$b in the triangle below. Give your answer correct to two decimal places.
Use Pythagoras' theorem to determine whether this is a right-angled triangle. 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 No
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.