Find the 13

Find the equation of the circle inscribed in the rhombus ABCD where A[1, -2], B[8, -3] and C[9, 4].

Result

e = (Correct answer is: )

Solution:

$$\begin{gathered} A[1,-2];B[8,-3];C[9,4]\mathrm{.} \\ S=AC\mathrm{/}2 \\ {x}_0=(9-1)\mathrm{/}2=4 \\ {y}_0=(4-(-2))\mathrm{/}2=3 \\ AB: \\ a=1 \\ a\cdot 1+b\cdot (-2)+c=0 \\ a\cdot 8+b\cdot (-3)+c=0 \\ 1\cdot 1+b\cdot (-2)+c=0 \\ 1\cdot 8+b\cdot (-3)+c=0 \\ 2b-c=1 \\ 3b-c=8 \\ b=7 \\ c=13 \\ v=\sqrt{a^2+b^2}=\sqrt{1^2+7^2}=5\sqrt{2}\doteq 7.0711 \\ r={\displaystyle \frac{a\cdot x_0+b\cdot y_0+c}{v}}={\displaystyle \frac{1\cdot 4+7\cdot 3+13}{7.0711}}\doteq 5.374 \\ R=r^2=5.374^2={\displaystyle \frac{722}{25}}=28.88 \\ e:(x-x_0{)}^2+(y-y_0{)}^2=r^2 \\ e:(x-4{)}^2+(y-3{)}^2=28.88 \end{gathered}$$

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