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Homework Statement
$AB = 20 cm$, $m∠A = 30°$ , and $m∠C = 45°$ . Express the number of centimeters in the length of $BC$ in simplest radical form.
Homework Equations
$sin A = sin C$
The Attempt at a Solution
$AB = 20, BC = x$
D is the point where this obtuse triangle separates into 2 right triangles
$BD/20 = sin A$
$AD/20 = cos A$
30-60-90 triangle
$1:2:\sqrt{3}$
BD is 10 according to this ratio which means that sin A is 1/2 and AD would be $20\sqrt{3}$
sin C is the same but for a 45-45-90 triangle instead.
45-45-90 triangle
$1:1:\sqrt{2}$
But here is where I am stuck. I am trying to find the side lengths of the 45-45-90 triangle with the trigonometric ratios being the same for both triangles but the angles being different so that I know the hypotenuse BC. But I don't know what side lengths will give me the trigonometric ratios being the same and the $1:2:\sqrt{3}$ and $1:1:\sqrt{2}$ side length ratios being true.