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?

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?

• A. 20
• B. 24
• C. 23
• D. 30

Answer: (C)

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

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