# Rectangular triangle

The lengths of the rectangular triangle sides with a longer leg 12 cm form an arithmetic sequence. What is the area of the triangle?

Result

S =  54 cm2

#### Solution:

$a=b-x \ \\ b=12 \ \text{cm} \ \\ c=b+x \ \\ \ \\ c^2=a^2 + b^2 \ \\ (12+x)^2=(12-x)^2 + 12^2 \ \\ 48x=144 \ \\ x=144/48=3 \ \\ \ \\ a=b-x=12-3=9 \ \text{cm} \ \\ c=b+x=12+3=15 \ \text{cm} \ \\ \ \\ S=\dfrac{ a \cdot \ b }{ 2 }=\dfrac{ 9 \cdot \ 12 }{ 2 }=54 \ \text{cm}^2$

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