Q1:

Choose the correct option and give justification. From a point $Q$, the length of the tangent to a circle is $24$ cm and the distance of $Q$ from the centre is $25$ cm. The radius of the circle is

Given:

A circle with center $O$. A point $Q$ lies outside the circle such that the distance from the center $O$ to point $Q$ is $OQ = 25$ cm. A tangent is drawn from $Q$ to the circle, touching the circle at point $P$, such that the length of the tangent $QP = 24$ cm.

To Find:

The radius of the circle, denoted as $OP$.

Step 1: Applying the Tangent-Radius Theorem

According to the theorem: "The tangent at any point of a circle is perpendicular to the radius through the point of contact."

Therefore, $OP \perp QP$. This implies that $\angle OPQ = 90^\circ$.

Step 2: Identifying the Triangle

Since $\angle OPQ = 90^\circ$, the triangle $\triangle OPQ$ is a right-angled triangle, where $OQ$ is the hypotenuse, and $OP$ and $QP$ are the legs of the triangle.

Step 3: Applying the Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. [Pythagorean Theorem: $a^2 + b^2 = c^2$]

Substituting the known values into the equation:

$OP^2 + QP^2 = OQ^2$

$OP^2 + (24)^2 = (25)^2$

Step 4: Algebraic Calculation

Calculate the squares of the given lengths:

$OP^2 + 576 = 625$

Isolate $OP^2$ by subtracting 576 from both sides:

$OP^2 = 625 - 576$

$OP^2 = 49$

Take the square root of both sides:

$OP = \sqrt{49}$

$OP = 7$ cm

Conclusion:

The radius of the circle $OP$ is $7$ cm.

Final Answer: 7 cm