Isosceles right triangles (45-45-90 special right triangles)

While the Pythagorean Theorem can apply to any kind of right-angled triangle, there are particular types of right-angled triangles whose side lengths and angles have helpful properties. 

The first kind of triangle that we want to look at is the $45^\circ-45^\circ-90^\circ$45°−45°−90° triangle. 

The first thing to notice about this kind of triangle is that it has two congruent angles and is, therefore, an isosceles triangle. This, in turn, means that the two sides of the triangle that are not the hypotenuse are equal in length. That is, $a=b$a=b in the diagram above.

Knowing this fact means that we can also use the Pythagorean Theorem to find out what the length of the hypotenuse should be. For example, let's look at the case where $a=b=1$a=b=1.

We can then find the value of $c$c in the following way.

$1^2+1^2$12+12	$=$=	$c^2$c2	Using the Pythagorean Theorem
$1+1$1+1	$=$=	$c^2$c2	Simplifying the powers
$c^2$c2	$=$=	$2$2	Evaluating the addition
$c$c	$=$=	$\sqrt{2}$√2	Taking the positive square root of both sides

So, here we have a $45^\circ-45^\circ-90^\circ$45°−45°−90° with all side lengths filled in:

Note that any triangle with angle measures of $45^\circ$45°, $45^\circ$45° and $90^\circ$90° is necessarily similar to this triangle. Using what we know about similar triangles, this means that the sides of any $45^\circ-45^\circ-90^\circ$45°−45°−90° triangle are in the ratio $1:1:\sqrt{2}$1:1:√2.

 

 

Practice questions

Question 1

Question 2

Question 3