Triangle arithmetic sides

The lengths of the sides of a right triangle form the first 3 terms of the arithmetic sequence. Its area is 6 cm². Find length of its sides.

Final Answer:

a =  3
b =  4
c =  5

Step-by-step explanation:

$$\begin{gathered} a,b,c \\ S=6{\text{cm}}^2 \\ S=\frac{ab}{2} \\ b=a+d \\ c=a+2d \\ c=\sqrt{a^2+b^2}=\sqrt{a^2+(a+d{)}^2} \\ (a+2d{)}^2=a^2+(a+d{)}^2 \\ a\cdot (a+d)=2\cdot S=12 \\ {a}^2+4a+4d^2=a^2+a^2+2ad+d^2 \\ {a}^2+ad=12 \\ 4a+3d^2=a^2+2ad \\ {a}^2+ad=12 \\ 4a+3d^2=a^2+2(12-a^2) \\ 4a+3d^2=a^2+24-2a^2 \\ a=3 \\ d=1 \end{gathered}$$

$b=a+d=3+1=4$

$c=b+d=4+1=5$

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