Question

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

Answer 1

Steps of construction:

1. Draw a circle of radius $4cm$ at center $C$.

2. Draw another circle of radius $6cm$ at the same center $C$.

3. Mark a point $P$ on the larger circle.

4. Join $PC$.

5. Draw a perpendicular bisector of $PC$, which cuts $PC$ at the point $M$.

6. Draw a circle with radius $PM=CM$, which cuts the smaller circle at $A$ and $B$ respectively.

7. Join $PA$ and $PB$.

Hence, $PA$ and $PB$ are the required tangents.

By measuring the lengths of these tangents, will get length of each tangent is $4.47cm$.

Finding the lengths of the tangents $PA$ and $PB$:

Since $△CAP$ is a right angled triangle with $\angle A={90}^{\circ }$

By using Pythagoras therem,

$C{A}^2+P{A}^2=C{P}^2$

$\Rightarrow P{A}^2=C{P}^2-C{A}^2$

Here, $CP=6cm$ [Since, the radius of the larger circle]

$CA=3cm$ [Since, the radius of the smaller circle]

$\Rightarrow P{A}^2=(6cm{)}^2-(4cm{)}^2$

$\Rightarrow P{A}^2=36c{m}^2-16c{m}^2$

$\Rightarrow P{A}^2=20c{m}^2$

$\Rightarrow PA=2\sqrt{5}cm$

$\Rightarrow PA=2\times 2.236cm$ [ Since, $\sqrt{5}=2.236$ Approximately]

$\Rightarrow PA=4.47cm$ [Approximately]

Since, the lengths of the tangents drawn from an external point to the circle are equal.

$\Rightarrow PA=PB=4.47cm$.

Hence, verified.