Right-angled triangle

Determine the area of a right triangle whose side lengths form successive members of an arithmetic progression, and the radius of the circle described by the triangle is 5 cm.

Final Answer:

S =  24 cm²

Step-by-step explanation:

$$\begin{gathered} r=5\text{cm} \\ c=2\cdot r=2\cdot 5=10\text{cm} \\ {c}^2=a^2+b^2 \\ a=c-2d \\ b=c-d \\ {c}^2=(c-2d{)}^2+(c-d{)}^2 \\ 10^2=(10-2d{)}^2+(10-d{)}^2 \\ -5d^2+60d-100=0 \\ 5d^2-60d+100=0 \\ 5...\text{ prime number} \\ 60=2^2\cdot 3\cdot 5 \\ 100=2^2\cdot 5^2 \\ \text{GCD}(5,60,100)=5 \\ {d}^2-12d+20=0 \\ a=1;b=-12;c=20 \\ D=b^2-4ac=12^2-4\cdot 1\cdot 20=64 \\ D>0 \\ {d}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{12\pm \sqrt{64}}{2} \\ {d}_{1,2}=\frac{12\pm 8}{2} \\ {d}_{1,2}=6\pm 4 \\ {d}_1=10 \\ {d}_2=2 \\ d<c \\ d=d_2=2 \\ b=c-d=10-2=8\text{cm} \\ a=c-2\cdot d=10-2\cdot 2=6\text{cm} \\ S=a\cdot b/2=6\cdot 8/2=24{\text{cm}}^2 \end{gathered}$$

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