The area of the triangle is 84, one of its sides is 13, and the radius of the inscribed circle is 4. Find the other two sides of the triangle.

Let a triangle ABC be given. The area of the triangle S (ABC) = 84. One of its sides is equal to AB = 13, and the radius of the inscribed circle is r = 4. To find the other two sides of the triangle, we denote the lengths of the sides of the triangle ABC by the letters a, b and c, the semiperimeter p = (a + b + c): 2, then according to Heron’s formula S (ABC) ^ 2 = p ∙ (p – a) ∙ (p – b) ∙ (p – c). On the other hand, S (ABC) = r ∙ p. Let us express p = S (ABC): r or p = 84: 4 = 21, then (a + b + 13): 2 = 21; b = 29 – a.
S (ABC) ^ 2 = 21 ∙ (21 – a) ∙ (21 – (29 – a)) ∙ (21 – 13); a ^ 2 – 29 ∙ a + 210 = 0: a1 = 14, a2 = 15, then b1 = 15 and b2 = 14.
Answer: The other two sides of the triangle are 14 and 15.