Question

find the area of the triangle abc.
a = 104.1 m b = 82.7 m c = 98.6 m
what is the area of the triangle?
□ m² (round to the nearest square meter as needed.)

Explanation:

Step1: Calculate the semi - perimeter (s)

The formula for the semi - perimeter \(s\) of a triangle with sides \(a\), \(b\), and \(c\) is \(s=\frac{a + b + c}{2}\).
Given \(a = 104.1\space m\), \(b=82.7\space m\), \(c = 98.6\space m\).
\(s=\frac{104.1+82.7 + 98.6}{2}=\frac{285.4}{2}=142.7\space m\)

Step2: Use Heron's formula to find the area (A)

Heron's formula for the area of a triangle is \(A=\sqrt{s(s - a)(s - b)(s - c)}\)
Substitute \(s = 142.7\), \(a = 104.1\), \(b = 82.7\), \(c=98.6\) into the formula:
First, calculate \(s - a=142.7-104.1 = 38.6\)
\(s - b=142.7 - 82.7=60\)
\(s - c=142.7-98.6 = 44.1\)
Then, \(A=\sqrt{142.7\times38.6\times60\times44.1}\)
Calculate \(142.7\times38.6=142.7\times(30 + 8+0.6)=142.7\times30+142.7\times8 + 142.7\times0.6=4281+1141.6+85.62 = 5508.22\)
\(5508.22\times60 = 330493.2\)
\(330493.2\times44.1=330493.2\times(40 + 4+0.1)=330493.2\times40+330493.2\times4+330493.2\times0.1=13219728+1321972.8 + 33049.32=14574750.12\)
\(A=\sqrt{14574750.12}\approx3817.7\approx3818\)

Answer:

\(3818\space m^{2}\)