I have been revising some maths equations and see that you can work out the third side of an isosceles triangle using the formula $\sqrt2 x$
$x$ being one of the equal sides.
Could someone explain how this works?
I understand the theorem:
$$a^2+b^2=c^2$$
But, I am struggling to understand the relationship between the two formulas.
Thank you in advance :)
$\endgroup$ 32 Answers
$\begingroup$If $$a^2 + b^2 = c^2$$
and $a=b$:
$$a^2 + a^2 = 2 a^2 = c^2$$
You can indeed find the third side of the right triangle with the following formula:
$$\sqrt{2 a^2} = \sqrt{c^2}$$ $$\sqrt{2} a = c$$
Both sides having the same length allows you to turn the sum into a product, which you can partially calculate the square root of. (or at least calculate the root and be able to write down the result in a finite amount of time)
If the angle between the 2 sides of equal length is not 90°, you can start from the general formula for the third side, which also includes the angle $\alpha$ between them:
$$a^2 + b^2 -2ab\cos (\alpha)= c^2$$
again $a=b$:
$$a^2 + a^2 -2aa\cos (\alpha)=2a^2(1-\cos (\alpha))= c^2$$
unfortunately, the angle is still there and the formula does not simplify as much as it does for the special case of 90°.
$\endgroup$ 0 $\begingroup$$$a^2 + b^2 = c^2 $$
$$a = b = 1$$
$$(1)^2 + (1)^2 = 2$$
$$ c = \sqrt2$$
I simply plugged in $1$ for $ a $ and $ b $ so that the relationship would show best. $c = \sqrt2*(1)$
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