Question

13 multiple choice 1 point find the area of the triangle. a = 20.8 b = 15.2 c = 13.8 113.84 104.84 107.84

Explanation:

Step1: Calculate the semi - perimeter \(s\)

The formula for the semi - perimeter of a triangle with sides \(a\), \(b\), and \(c\) is \(s=\frac{a + b + c}{2}\).
Substitute \(a = 20.8\), \(b = 15.2\), and \(c = 13.8\) into the formula:
\(s=\frac{20.8+15.2 + 13.8}{2}=\frac{49.8}{2}=24.9\)

Step2: Use Heron's formula \(A=\sqrt{s(s - a)(s - b)(s - c)}\)

Substitute \(s = 24.9\), \(a = 20.8\), \(b = 15.2\), and \(c = 13.8\) into Heron's formula:
\(A=\sqrt{24.9(24.9 - 20.8)(24.9 - 15.2)(24.9 - 13.8)}\)
\(=\sqrt{24.9\times4.1\times9.7\times11.1}\)
\(=\sqrt{24.9\times4.1\times(9.7\times11.1)}\)
\(=\sqrt{24.9\times4.1\times107.67}\)
\(=\sqrt{(24.9\times4.1)\times107.67}\)
\(=\sqrt{102.09\times107.67}\)
\(\approx\sqrt{11000}\) (approximate calculation)
Another way:
We can also use the formula \(A=\frac{1}{2}ab\sin C\), but since no angle is given, Heron's formula is more straightforward.
Let's recalculate Heron's formula accurately:
\(s = 24.9\), \(s - a=24.9 - 20.8 = 4.1\), \(s - b=24.9 - 15.2 = 9.7\), \(s - c=24.9 - 13.8 = 11.1\)
\(A=\sqrt{24.9\times4.1\times9.7\times11.1}\)
\(24.9\times4.1 = 102.09\), \(9.7\times11.1=107.67\)
\(102.09\times107.67=(100 + 2.09)\times(100+7.67)\)
\(=100\times100+100\times7.67+2.09\times100+2.09\times7.67\)
\(=10000 + 767+209+16.0303\)
\(=10992.0303\)
\(A=\sqrt{10992.0303}\approx104.84\)

Answer:

104.84