What is the sum of the lengths of the sides of a triangle with sides of 4, 15, and one other side that creates an area of 13.15 square units?

To find the sum of the lengths of the sides of the triangle, we begin by recognizing that two sides of the triangle are given as 4 and 15 units. We need to find the length of the third side, which we can denote as ( x ), and it must satisfy the condition that the area of the triangle equals 13.15 square units.

Using the formula for the area of a triangle given two sides and the included angle, we can express the area ( A ) as follows:

[

A = \frac{1}{2} \times base \times height

]

In this problem, we can also utilize Heron's formula, which helps in finding the area based on the sides of the triangle. First, we need to determine the semi-perimeter ( s ):

[

s = \frac{4 + 15 + x}{2} = \frac{19 + x}{2}

]

Then, using Heron's formula for the area:

[

A = \sqrt{s(s - a)(s - b)(s - c)}

]

Where ( a = 4 ), ( b = 15 ), and ( c = x ). Substituting into Heron's