RT and circles

Solve right triangle if the radius of inscribed circle is r=9 and radius of circumscribed circle is R=23.

Correct answer:

a =  37.83
b =  26.17
c =  46

Step-by-step explanation:

$$\begin{gathered} R={\displaystyle \frac{c}{2}} \\ c=2R=46 \\ r={\displaystyle \frac{a+b-c}{2}} \\ a+b=64 \\ {a}^2+b^2=2116 \\ 2a^2-128a+1980=0 \\ p=2;q=-128;r=1980 \\ D=q^2-4pr=128^2-4\cdot 2\cdot 1980=544 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{128\pm \sqrt{544}}{4}}={\displaystyle \frac{128\pm 4\sqrt{34}}{4}} \\ {a}_{1,2}=32\pm 5.8309518948453 \\ {a}_1=37.830951894845 \\ {a}_2=26.169048105155 \\ \text{ Factored form of the equation: } \\ 2(a-37.830951894845)(a-26.169048105155)=0 \end{gathered}$$

$b=(-(-128)-(23.323807579381))\mathrm{/}(2\cdot (2))=26.17$

$c=2\cdot 23=46$

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