Question

Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.

Answer 1

Steps of Construction:
Step I: With O as a centre and radius equal to 4 cm, a circle is drawn.

Step II: With O as a centre and radius equal to 6 cm, a concentric circle is drawn.
Step III: P be any point on the circle of radius 6 cm and OP is joined.
Step IV: Perpendicular bisector of OP is drawn which cuts it at M.
Step V: With M as a centre and OM as a radius, a circle is drawn which intersect the circle of radius 4 cm at Q and R.
Step VI: PQ and PR are joined.
Thus, PQ and PR are the two tangents.
Measurement:
OQ = 4 cm (Radius of the smaller circle)
PO = 6 cm ( Radius of the larger circle)
$\angle PQO={90}^{\circ }$ (Angle in the semi circle)
Applying Pythagoras theorem in $\Delta PQO$,
$P{Q}^2+Q{O}^2=P{O}^2$
$\Rightarrow P{Q}^2+4^2=6^2$
$\Rightarrow P{Q}^2+16=36$
$\Rightarrow P{Q}^2=36-16$
$\Rightarrow P{Q}^2=20$
$\Rightarrow PQ=2\sqrt{5}cm$
Justification:
$\angle PQO={90}^{\circ }$ (Angle in the semi circle)
$\therefore OQ\perp PQ$
Therefore, OQ is the radius of the circle then PQ has to be a tangent of the circle.
Similarly, PR is a tangent of the circle.