# How long is side $a$ in this right triangle if $a = b$, given $c$?

A right angled triangle has side lengths labeled as so.

A common geometric construction that shows three squares sitting upon the sides of a right triangle with lengths A, B, and C

However unlike in this diagram $a = b$.

How can $a$ be calculated given $c$?

Would $a = c \cdot d$ where $d$ is a constant?

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A suggestion: the use of $N$ as a variable is prone to make people think it is a natural number, so it should be avoided. $D$ would be a better choice. – Ross Millikan Mar 15 '11 at 2:22

We know that $c^2 = a^2+b^2$ from Pythagorean Theorem or $c^2 = 2a^2$. Thus $c = a \sqrt{2}$.
... and so $a = \frac{c}{\sqrt{2}}$, making $N = \frac{1}{\sqrt{2}} \approx 0.7071$ – Henry Mar 15 '11 at 8:44
If A=B , then the angles of the triangle are 45:45:90, And as per the 45:45:90 theorem, the side opposite the 45 degree angle is hypotenuse/√2
So in this case value of A will be C/√2