Right-angled triangle

Determine the content 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.

Correct answer:

S =  24 cm²

Step-by-step explanation:

$$\begin{gathered} r=5\text{ cm } \\ c=2\cdot r=2\cdot 5=10\text{ cm } \\ {c}^2=a^2+b^2 \\ a=c-2d \\ b=c-d \\ {c}^2=(c-2d{)}^2+(c-d{)}^2 \\ 10^2=(10-2d{)}^2+(10-d{)}^2 \\ -5d^2+60d-100=0 \\ 5d^2-60d+100=0 \\ a=5;b=-60;c=100 \\ D=b^2-4ac=60^2-4\cdot 5\cdot 100=1600 \\ D>0 \\ {d}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{60\pm \sqrt{1600}}{10}} \\ {d}_{1,2}={\displaystyle \frac{60\pm 40}{10}} \\ {d}_{1,2}=6\pm 4 \\ {d}_1=10 \\ {d}_2=2 \\ \text{ Factored form of the equation: } \\ 5(d-10)(d-2)=0 \\ d<c \\ d=d_2=2 \\ b=c-d=10-2=8\text{ cm } \\ a=c-2\cdot d=10-2\cdot 2=6\text{ cm } \\ S=a\cdot b\mathrm{/}2=6\cdot 8\mathrm{/}2=24{\text{cm}}^2 \end{gathered}$$

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