Touch circle

Point A has a distance (A, k) = 10 cm from a circle k with radius r = 4 cm and center S.
Calculate:
a) the distance of point A from the point of contact T if the tangent to the circle is drawn from point A
b) the distance of the contact point T from the line SA

Correct answer:

x =  14.5602 cm
v =  4.1601 cm

Step-by-step explanation:

$$\begin{gathered} r=4\text{cm} \\ a=10\text{cm} \\ {x}^2=r^2+(r+a{)}^2 \\ x=\sqrt{r^2+(r+a{)}^2}=\sqrt{4^2+(4+10{)}^2}=2\sqrt{53}=14.5602\text{cm} \end{gathered}$$

$$\begin{gathered} S=x\cdot r/2=14.5602\cdot 4/2=4\sqrt{53}{\text{cm}}^2\doteq 29.1204{\text{cm}}^2 \\ S=(r+a)\cdot v/2 \\ v=2\cdot S/(r+a)=2\cdot 29.1204/(4+10)=4.1601\text{cm} \end{gathered}$$

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