Find the length of the third side of a triangle if the area

Find the length of the third side of a triangle if the area of the triangle is 18 and two of its sides have lengths of 5 and 10.??

Solution

The formula for the area using two sides and the internal angle they make, may be written as follows

18 = (1/2) * 5 * 10*sin(A)

which gives: sin(A) = 18/25

We now use the cosine formula to fin the length x of the third side opposing angle A as follows:

x² = 5² + 10² - 2*5*10*cos(A)

with cos(A) = sqrt(1 - sin(A)²)

Substitute in the expression for x² and solve for x to obtain x = 7.46 (approximated to 3 significant digits)