In the given figure, O is the centre of two concentric circles of radii 4 cm and 6 cm respectively. PA and PB are tangents to the outer and inner circles, respectively. If PA = 10 cm, find the length of PB up to one decimal place.
Given, $O$ is the centre of two concentric circles of radii $O A=6 \mathrm{~cm}$ and $O B=4 \mathrm{~cm}$.
$P A$ and $P B$ are the two tangents to the outer and inner circles respectively and $P A=10 \mathrm{~cm}$.
Now, tangent drawn from an external point is perpendicular to the radius at the point of contact.
$\therefore \angle O A P=\angle O B P=90^{\circ}$
$\therefore$ From right-angled $\triangle O A P, O P^{2}=O A^{2}+P A^{2}$
$=>O P=\sqrt{O A^{2}+P A^{2}}$
$=>O P=\sqrt{6^{2}+10^{2}}$
$=>O P=\sqrt{136} \mathrm{~cm}$
$\therefore$ From right-angled $\triangle O A P, O P^{2}=O B^{2}+P B^{2}$
$=>P B=\sqrt{O P^{2}-O B^{2}}$
$=>P B=\sqrt{136-16}$
$=>P B=\sqrt{120} \mathrm{~cm}$
$=>P B=10.9 \mathrm{~cm} .$
$\therefore$ The length of $P B$ is $10.9 \mathrm{~cm}$.
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