Question 1073094
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If the area of traingle is 80 sq unit.the longest side is 20 unit and other side is 10 unit then the third side is what??
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<pre>
The area of a triangle (of any triangle) is half the product of two sides lengths and the sine of the angle between them:

S = {{{(1/2)*a*b*sin(alpha)}}}.


Substituting the given data, you obtain {{{sin(alpha)}}} = {{{(2*S)/(a*b)}}} = {{{(2*80)/(20*10)}}} = {{{160/200}}} = {{{4/5}}}.


Then {{{cos(alpha)}}} = +/- {{{sqrt(1-sin^2(alpha))}}} = +/- {{{3/5}}}.


{{{cos(alpha)}}} can not be negative under the given condition.

Indeed, otherwise the angle {{{alpha}}} is obtuse and our 20 unit side is not the longest side.

So, {{{cos(alpha)}}} = {{{3/5}}}.

Now apply the Cosine Law theorem. You will get for the length of the third side

{{{c^2}}} = {{{a^2 + b^2 - 2ab*cos(alpha)}}} = {{{10^2 + 20^2 - 2*10*20*(3/5)}}} = {{{500 - 2*10*4*3}}} = {{{500 - 240}}} = 260.

Thus  c = {{{sqrt(260)}}} = 16.125 units (approximately).
</pre>

Solved.