Final answer:
The area of the triangle can be calculated using the formula A = (30 - b - c) * (88 / π), where b and c are the lengths of the two sides of the triangle.
Step-by-step explanation:
The perimeter of a triangle is the sum of the lengths of its three sides. The formula for the perimeter of a triangle is P = a + b + c, where a, b, and c are the lengths of the three sides of the triangle. In this case, the perimeter of the triangle is given as 30 cm.
The circumference of the incircle is the distance around the circle formed by the incenter of the triangle. The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle. In this case, the circumference of the incircle is given as 88 cm.
Let's denote the lengths of the three sides of the triangle as a, b, and c. Then we have:
a + b + c = 30
Let's denote the radius of the incircle as r. Then we have:
2πr = 88
From the first equation, we can solve for one of the side lengths in terms of the other two side lengths:
a = 30 - b - c
Substituting this into the second equation, we get:
2πr = 88
Let's solve for r:
r = 88 / (2π)
Now, we can substitute the value of r into the equation for a:
a = 30 - b - c
Substituting the value of r, we get:
a = 30 - b - c
Now, we can use the formula for the area of a triangle, which is A = 1/2 * base * height. The base of the triangle is one of the side lengths, and the height can be found by drawing a perpendicular line from one of the vertices of the triangle to the opposite side. Let's denote the base of the triangle as a:
A = 1/2 * a * h
Substituting the value of a, we get:
A = 1/2 * (30 - b - c) * h
Now, we can substitute the value of h into the equation for the area:
A = 1/2 * (30 - b - c) * (2 * r)
Substituting the values of r and h, we get:
A = 1/2 * (30 - b - c) * (2 * (88 / (2π)))
Now, we can simplify the expression:
A = (30 - b - c) * (88 / π)
Therefore, the area of the triangle is (30 - b - c) * (88 / π).