Two chords

Calculate the length of chord AB and perpendicular chord BC to circle if AB is 4 cm from the center of the circle and BC 8 cm from the center of the circle.

Correct result:

a =  16 cm
b =  8 cm

Solution:

$$\begin{gathered} h_1=4\text{ cm } \\ {h}_2=8\text{ cm } \\ {r}^2=h_1^2+h_2^2 \\ r=\sqrt{h_1^2+h_2^2}=\sqrt{4^2+8^2}=4\sqrt{5}\text{ cm}\doteq 8.9443\text{ cm } \\ D=2\cdot r=2\cdot 8.9443=8\sqrt{5}\text{ cm}\doteq 17.8885\text{ cm } \\ a<D,b<D \\ {r}^2=h_1^2+(a\mathrm{/}2{)}^2 \\ a=2\cdot \sqrt{r^2-h_1^2}=2\cdot \sqrt{8.9443^2-4^2}=16\text{ cm} \end{gathered}$$

$$\begin{gathered} r^2=h_2^2+(b\mathrm{/}2{)}^2 \\ b=2\cdot \sqrt{r^2-h_2^2}=2\cdot \sqrt{8.9443^2-8^2}=8\text{ cm} \end{gathered}$$

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