Step-by-step Construction
- Take a point \(O\) on your paper. This will be the common centre.
- Using a compass, draw a circle with radius \(3\,\text{cm}\). This is the inner circle.
- From the same centre \(O\), draw another circle with radius \(5\,\text{cm}\). This is the outer circle. Now you have two concentric circles.
- Mark a point \(T\) anywhere on the outer circle (radius \(5\,\text{cm}\)).
- Join the line \(OT\).
- Now, to construct the tangents from \(T\) to the inner circle:
- Draw the circle with diameter \(OT\).
- This new circle will intersect the inner circle (radius \(3\,\text{cm}\)) at two points. Let those points be \(A\) and \(B\).
- Join \(TA\) and \(TB\). These are the required tangents.
Step-by-step Verification
We know that a radius drawn to the point of contact is perpendicular to the tangent.
So, in triangle \(OTA\):
- \(OA = 3\,\text{cm}\) (radius of inner circle)
- \(OT = 5\,\text{cm}\) (radius of outer circle)
- \(\angle OAT = 90^\circ\)
By Pythagoras theorem:
\[
TA^2 = OT^2 - OA^2 = (5)^2 - (3)^2 = 25 - 9 = 16
\]
\[
TA = \sqrt{16} = 4\,\text{cm}
\]
Hence, the length of each tangent is \(4\,\text{cm}\).