Q2:

In each of the following, give also the justification of the construction: 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.

Given: Two concentric circles with center $O$. The radius of the inner circle is $r_1 = 4\text{ cm}$ and the radius of the outer circle is $r_2 = 6\text{ cm}$.

To Find: Construct a tangent from a point $P$ on the outer circle to the inner circle, measure its length, and verify the result using the Pythagorean theorem.

Step 1: Construction Procedure

1. Draw a circle with center $O$ and radius $4\text{ cm}$.

2. Draw a concentric circle with center $O$ and radius $6\text{ cm}$.

3. Take a point $P$ on the outer circle. Join $OP$.

4. Find the midpoint $M$ of $OP$ by drawing the perpendicular bisector of $OP$.

5. With $M$ as the center and $MO$ as the radius, draw a circle. Let this circle intersect the inner circle at points $T$ and $T'$.

6. Join $PT$ and $PT'$. These are the required tangents.

Step 2: Justification

Join $OT$. Since $OT$ is the radius of the inner circle and $PT$ is the tangent, $\angle OTP = 90^\circ$ [Since the tangent at any point of a circle is perpendicular to the radius through the point of contact].

In $\triangle OTP$, by the Pythagorean theorem:

$OP^2 = OT^2 + PT^2$

Step 3: Calculation and Verification

Given $OP = 6\text{ cm}$ (radius of the outer circle) and $OT = 4\text{ cm}$ (radius of the inner circle).

Substituting the values into the Pythagorean equation:

$6^2 = 4^2 + PT^2$

$36 = 16 + PT^2$

$PT^2 = 36 - 16$

$PT^2 = 20$

$PT = \sqrt{20} = 2\sqrt{5}\text{ cm}$

Using $\sqrt{5} \approx 2.236$:

$PT \approx 2 \times 2.236 = 4.47\text{ cm}$

Final Answer: The length of the tangent is $2\sqrt{5}\text{ cm}$ or approximately $4.47\text{ cm}$.