Draw a line segment $AB$ of length $8 \, cm$. Taking $A$ as centre,draw a circle of radius $4 \, cm$ and taking $B$ as centre,draw another circle of radius $3 \, cm$. Construct tangents to each circle from the centre of the other circle. Also,provide the justification for the construction.

(N/A) The tangents can be constructed on the given circles as follows:
$1.$ Draw a line segment $AB$ of $8 \, cm$. Taking $A$ and $B$ as centres,draw two circles of $4 \, cm$ and $3 \, cm$ radius respectively.
$2.$ Bisect the line $AB$. Let the mid-point of $AB$ be $C$. Taking $C$ as centre,draw a circle of radius $AC$ (or $BC$),which will intersect the circles at points $P, Q, R,$ and $S$. Join $BP, BQ, AS,$ and $AR$. These are the required tangents.
Justification:
The construction can be justified by proving that $AS$ and $AR$ are the tangents to the circle (whose centre is $B$ and radius is $3 \, cm$) and $BP$ and $BQ$ are the tangents to the circle (whose centre is $A$ and radius is $4 \, cm$). For this,join $AP, AQ, BS,$ and $BR$.
$\angle ASB$ is an angle in the semi-circle. We know that an angle in a semi-circle is a right angle.
$\therefore \angle ASB = 90^{\circ}$
$\Rightarrow BS \perp AS$
Since $BS$ is the radius of the circle,$AS$ must be a tangent to the circle. Similarly,$AR, BP,$ and $BQ$ are the tangents.