Find the length of a tangent from a point $Q$ at a distance of $25\text{ cm}$ from the center, given that the radius of the circle is $7\text{ cm}$ (or $24\text{ cm}$ tangent length with $25\text{ cm}$ distance).

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Step-by-Step Solution

Step 1: Understand the Geometry

We are given a circle with center $O$ and a point $Q$ outside the circle. A tangent is drawn from $Q$ to the circle, touching it at point $T$. The radius of the circle is $OT = 7\text{ cm}$, and the distance from the center $O$ to point $Q$ is $OQ = 25\text{ cm}$. We need to find the length of the tangent $QT.

Step 2: Apply Tangent-Radius Theorem

According to the tangent-radius theorem, the radius drawn to the point of tangency is perpendicular to the tangent. Therefore, the angle $\angle OTQ$ is $90^\circ$, making $\triangle OTQ$ a right-angled triangle.

Step 3: Use Pythagorean Theorem

Since $\triangle OTQ$ is a right-angled triangle with the right angle at $T$, we can apply the Pythagorean theorem. The hypotenuse is $OQ$, and the other two sides are $OT$ and $QT.

Step 4: Substitute Values and Solve for QT

Substitute the given values $OQ = 25\text{ cm}$ and $OT = 7\text{ cm}$ into the Pythagorean theorem. Then, solve the equation for $QT$ by isolating $QT^2$ and taking the square root.