Question 733518
The measures of two angles of a triangle are {{{100^o}}} and {{{46^o}}} .
We know that the measures of all 3 angles add up to {{{180^o}}}, so the measure of the third angle is
{{{180^o-(100^o+46^o)=180^o-146^o=34^o}}}
In a triangle, the shortest side is opposite the smallest angle and the longest side is opposed the largest angle, so the shortest side is opposite the {{{34^o}}} angle.
The law of sines, says that in a triangle the ratio of the length of a side and the sine of the opposite angle is constant.
That means that the lengths, a, and b of the other two sides are related by
{{{12/sin(34^o)=a/sin(100^o)=b/sin(46^o)}}}
Using approximate values for the sines of those angles.
{{{12/0.559=a/0.985=b/0.719}}}
Solving the equations that can be split from there,
{{{12/0.559=a/0.985}}} --> {{{12/0.559*0.985=a}}} --> {{{highlight(a=21.1)}}} (rounded)
{{{12/0.559=b/0.719}}} --> {{{12/0.559*0.719=b}}} --> {{{highlight(b=15.4)}}} (rounded)