General triangle

Given a general triangle ABC. Its perimeter is 30 cm, with side a=2 cm longer than side b and 5 cm shorter than side c. Determine the area of the triangle.

Final Answer:

S =  26.8328 cm²

Step-by-step explanation:

$$\begin{gathered} o=30\text{cm} \\ a=2+b \\ a=c-5 \\ a+b+c=o \\ a+b+c=30 \\ a-b=2 \\ a-c=-5 \\ a+b+c=30 \\ Row2-Row1\to Row2 \\ b-c=-7 \\ Row3-Row1\to Row3 \\ 2b+c=28 \\ Pivot:Row2\leftrightarrow Row3 \\ Row3-\frac{1}{2}\cdot Row2\to Row3 \\ -1.5c=-21 \\ c=\frac{-21}{-1.5}=14 \\ b=\frac{28-c}{2}=\frac{28-14}{2}=7 \\ a=2+b=2+7=9 \\ a=9 \\ b=7 \\ c=14 \\ s=\frac{o}{2}=\frac{30}{2}=15\text{cm} \\ S=\sqrt{s\cdot (s-a)\cdot (s-b)\cdot (s-c)}=\sqrt{15\cdot (15-9)\cdot (15-7)\cdot (15-14)}=12\sqrt{5}=26.8328{\text{cm}}^2 \end{gathered}$$

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