Triangle circle calculation

The following elements are known in the right triangle ABC: a = 10 cm, height to side c h = 9.23 cm. Calculate o, R (radius of the inscribed circle), r (radius of the inscribed circle).

Final Answer:

o =  59.974 cm
r =  3.9995 cm
R =  12.9938 cm

Step-by-step explanation:

$$\begin{gathered} a=10\text{cm} \\ v=9.23\text{cm} \\ {c}_1=\sqrt{a^2-v^2}=\sqrt{10^2-9.23^2}\doteq 3.848\text{cm} \\ {v}^2=c_1{c}_2 \\ {c}_2=v^2/c_1=9.23^2/3.848\doteq 22.1395\text{cm} \\ c=c_1+c_2=3.848+22.1395\doteq 25.9875\text{cm} \\ b=\sqrt{c^2-a^2}=\sqrt{25.9875^2-10^2}\doteq 23.9865\text{cm} \\ o=a+b+c=10+23.9865+25.9875=59.974\text{cm} \end{gathered}$$

$$\begin{gathered} s=\frac{o}{2}=\frac{59.974}{2}\doteq 29.987\text{cm} \\ S=\frac{a\cdot b}{2}=\frac{10\cdot 23.9865}{2}\doteq 119.9324{\text{cm}}^2 \\ r=\frac{S}{s}=\frac{119.9324}{29.987}=3.9995\text{cm} \end{gathered}$$

$R=\frac{c}{2}=\frac{25.9875}{2}=12.9938\text{cm}$

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