We are given a circle centered at $O$, with a tangent at point $P$. The tangent is extended to point $Q$, and line segment $\overline{QS}$ intersects the circle at point $R$. We are tasked with finding the area of the circle.

Step 1: Analyze the diagram

The diagram provides the following key information:

• $OP$ is the radius of the circle, as $P$ lies on the circle and $OP$ is perpendicular to the tangent at $P$.
• $PQ = 8$, which is the length of the tangent from $P$ to $Q$.
• The segment $QS$ is extended and intersects the circle at $R$.
• The segment $OR = 3 + 4 = 7$, where the distances $OR$ are the sum of the two smaller segments $3$ and $4$.

Step 2: Apply the Power of a Point Theorem

The Power of a Point Theorem states that the power of point $Q$ with respect to the circle is equal to the product of the lengths of the segments of the line $\overline{QR}$ that intersects the circle. Specifically, the theorem gives:

$QP \cdot QS = OP^2$

From the problem:

• $PQ = 8$
• $QS = 3 + 4 = 7$
• The radius $OP = r$, which is what we need to find.

Step 3: Set up the equation

Using the Power of a Point Theorem, we substitute the known values:

$8 \cdot 7 = r^2$

This simplifies to:

$56 = r^2$

Step 4: Solve for $r$

To find the radius $r$, take the square root of both sides:

$r = \sqrt{56} = 2\sqrt{14}$

Step 5: Calculate the area of the circle

The area $A$ of a circle is given by the formula:

$A = \pi r^2$

Substitute $r^2 = 56$:

$A = \pi \times 56 = 56\pi$

Thus, the area of the circle is $\boxed{56\pi}$.

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If you'd like further details or have any questions, feel free to ask! Here are some related questions:

1. How does the Power of a Point Theorem work for tangents to a circle?
2. What is the geometric significance of the length of a tangent from a point outside the circle?
3. Can we apply the Power of a Point Theorem in other geometric configurations? Provide examples.
4. How do we calculate the area of a circle if we know the length of a chord and its perpendicular distance from the center?
5. What other properties of tangents to circles can help solve geometric problems?

Tip: In geometry, understanding the properties of tangents and secants to a circle can significantly simplify the solution process for many problems involving circles!