Question

Find x in this 45°-45°-90° triangle.

Answer 1

Know the relationships of the sides on a 45-45-90?

Answer 2

For any right triangle you have 2 legs and the hypotenuse. If the legs are a and b and the hypotenuse is c, then by the Pythagorean equation you can say:

\(a^2+b^2=c^2\)

However, on a 45-45-90, the legs are equal, so b can be replaced by a:

\(a^2+a^2=c^2\\[.5em]\)
\(2a^2=c^2\)

Now, lets solve this for c and a:

\(\sqrt{2a^2}=c\\[.5em]\)
\(a\sqrt{2}=c\)

and

\(a^2=\dfrac{c^2}{2}\\[.5em]\)
\(a=\sqrt{\dfrac{c^2}{2}}\\[.5em]\)
\(a=\dfrac{c}{\sqrt{2}}\)

Now, that is solved enough to use there, but typically it is rationalized:

\(a=\dfrac{c\sqrt{2}}{2}\)

OK. There you go. Two specific formulas, but you could just use:

\(2a^2=c^2\)

Whichever way you like to do it, all you need to do is put in what you are given and solve for what you need. If you are given a leg, you put it in for a and solve for the hypotenuse. If you are given a hypotenuse, put it in for c and solve for a leg.

Answer 3

Thank you!

Answer 4

np. Have fun!