Question

find the area of the triangle abc.
a = 103.6 m b = 76.8 m c = 78.8 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 \( a \), \( b \), \( c \) is \( s=\frac{a + b + c}{2}\).
Given \( a = 103.6\space m \), \( b = 76.8\space m \), \( c = 78.8\space m \)
\( s=\frac{103.6+76.8 + 78.8}{2}=\frac{259.2}{2}=129.6\space m \)

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

Heron's formula is \( A=\sqrt{s(s - a)(s - b)(s - c)} \)
Substitute \( s = 129.6 \), \( a = 103.6 \), \( b = 76.8 \), \( c = 78.8 \) into the formula:
\( s - a=129.6-103.6 = 26 \)
\( s - b=129.6 - 76.8=52.8 \)
\( s - c=129.6 - 78.8 = 50.8 \)
\( A=\sqrt{129.6\times26\times52.8\times50.8} \)
First, calculate \( 129.6\times26 = 3369.6 \)
Then, \( 52.8\times50.8=2682.24 \)
Next, \( 3369.6\times2682.24 = 3369.6\times2682.24\approx9038647.4496 \)
Then, \( A=\sqrt{9038647.4496}\approx3006.43\space m^{2}\approx3006\space m^{2} \)

Answer:

\( 3006 \)