Question

find the area of the triangle abc.
a = 342 cm b = 236 cm c = 424 cm
what is the area of the triangle?
\\( \square cm^{2} \\)
(simplify your answer. type an integer or decimal rounded t

Explanation:

Step1: Calculate the semi - perimeter

The semi - perimeter \(s=\frac{a + b + c}{2}\).
Substitute \(a = 342\), \(b = 236\), \(c = 424\) into the formula:
\(s=\frac{342+236 + 424}{2}=\frac{1002}{2}=501\)

Step2: Use Heron's formula

Heron's formula for the area of a triangle is \(A=\sqrt{s(s - a)(s - b)(s - c)}\)
Substitute \(s = 501\), \(a = 342\), \(b = 236\), \(c = 424\) into the formula:
\(A=\sqrt{501(501 - 342)(501 - 236)(501 - 424)}\)
\(=\sqrt{501\times159\times265\times77}\)
First, calculate \(501\times159 = 501\times(160-1)=501\times160-501=80160 - 501=79659\)
Then, \(79659\times265=(80000 - 341)\times265=80000\times265-341\times265=21200000-90365 = 21109635\)
Next, \(21109635\times77=(21109600+35)\times77=21109600\times77+35\times77=1625439200+2695=1625441895\)
\(A=\sqrt{1625441895}\approx40317.9\)

Answer:

\(40317.9\)