# Triangle with Angle,Side,Ratio known

#### xynon

Hello everybody,

I've got a triangle with 1 angle, 1 side and the ratio of the other 2 sides known. But no matter what I do, I keep running in circles.

Is there a way to leave the unknown "d" alone in one side of the equation?

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#### skeeter

MHF Helper
Pythagoras ...

$e^2 = (d+b)^2 + a^2$

Law of cosines ...

$e^2 = d^2+c^2 - 2dc\cos{E}$

therefore ...

$(d+b)^2 + a^2 = d^2+c^2 - 2dc\cos{E}$

$d^2+2db+b^2+a^2 = d^2+c^2 - 2dc\cos{E}$

$2db + 2dc\cos{E} = c^2 - (a^2+b^2)$

$d = \dfrac{c^2 - (a^2+b^2)}{2(b+c\cos{E})}$

#### DenisB

Alternatively, re-arrange the equation as a quadratic:

(z^2 - 1)d^2 - 2bd - a^2 - b^2 = 0

Solve for d.