Question

find the area of the triangle abc.
a = 102.3 m b = 89.1 m c = 101.7 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 of a triangle with sides \(a\), \(b\), \(c\) is \(s=\frac{a + b + c}{2}\).
Given \(a = 102.3\space m\), \(b=89.1\space m\), \(c = 101.7\space m\)
\(s=\frac{102.3+89.1 + 101.7}{2}=\frac{293.1}{2}=146.55\space m\)

Step2: Apply Heron's formula

Heron's formula for the area (\(A\)) of a triangle is \(A=\sqrt{s(s - a)(s - b)(s - c)}\)
Substitute \(s = 146.55\), \(a = 102.3\), \(b = 89.1\), \(c=101.7\) into the formula:
\(s - a=146.55-102.3 = 44.25\)
\(s - b=146.55 - 89.1=57.45\)
\(s - c=146.55-101.7 = 44.85\)
\(A=\sqrt{146.55\times44.25\times57.45\times44.85}\)
First, calculate the product inside the square root:
\(146.55\times44.25=146.55\times44 + 146.55\times0.25=6448.2+36.6375 = 6484.8375\)
\(57.45\times44.85=(57 + 0.45)\times(44+0.85)=57\times44+57\times0.85 + 0.45\times44+0.45\times0.85\)
\(=2508+48.45+19.8 + 0.3825=2508 + 48.45=2556.45+19.8=2576.25+0.3825 = 2576.6325\)
Then, \(6484.8375\times2576.6325\approx6484.8375\times2576.6325\)
\(6484.8375\times2576.6325=(6000 + 484.8375)\times(2500+76.6325)\)
\(=6000\times2500+6000\times76.6325+484.8375\times2500+484.8375\times76.6325\)
\(=15000000+459795+1212093.75+37150.546875\)
\(=15000000+459795 = 15459795+1212093.75=16671888.75+37150.546875 = 16709039.296875\)
\(A=\sqrt{16709039.296875}\approx4087.67\approx4088\)

Answer:

\(4088\space m^{2}\)