Construct a pair of tangent to …

Required: To 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 to verify the measurement by actual by actual calculation.
 
Steps of construction :

1. join PO and bisect it, Let M be the mid-point of PO.
2. Taking M as centre and MO as radius, draw a circle. Let it intersect the given circle at the point Q and R.
3. Join PQ
   Then PQ is the required tangent. By measurement, PQ = 4.5 cm By actual calculation, 
   {tex}\mathrm { PQ } = \sqrt { \mathrm { OP } ^ { 2 } - \mathrm { OQ } ^ { 2 } }{/tex} [By Pythagoras Theorem]
   {tex}= \sqrt { ( 6 ) ^ { 2 } - ( 4 ) ^ { 2 } }{/tex}
   {tex}= \sqrt { 36 - 16 } = \sqrt { 20 }{/tex}
   = 4.47 cm
   Justification: Join OQ. Then {tex}\angle \mathrm { PQO }{/tex} is an angle in the semicircle and, therefore,
   {tex}\angle P Q O = 90 ^ { \circ }{/tex}
   {tex}\Rightarrow \quad \text { PQ } \perp 0 Q{/tex}
   Since OQ is a radius of the given circle, PQ has to be a tangent to the circle.