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

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

Edit: Sorry PEV to be a pain and change the question, I mistyped it originally. Also I've changed N to D as per the comment about natural numbers. I had to edit the image link due to spam prevention for new users, I can't comment much below for the same reasons.

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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

## 2 Answers

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

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We know that $c^2 = a^2+b^2$ from Pythagorean Theorem or $c^2 = 2a^2$. Thus $c = a \sqrt{2}$.

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... and so $a = \frac{c}{\sqrt{2}}$, making $N = \frac{1}{\sqrt{2}} \approx 0.7071$ –  Henry Mar 15 '11 at 8:44