Question

From an external point $P$, two tangents $PA$ and $PB$ are drawn to a circle with centre $O$. At a point $E$ on the circle, a tangent is drawn which intersects $PA$ and $PB$ at $C$ and $D$ respectively. If $PA = 10$ cm, find the perimeter of $\triangle PCD$.

Answer 1

Step 1: Understanding the given situation:
We are given a circle with centre $O$. From an external point $P$, two tangents $PA$ and $PB$ are drawn to the circle, touching the circle at points $A$ and $B$ respectively. A tangent at point $E$ intersects $PA$ and $PB$ at points $C$ and $D$ respectively.
We are given that $PA = 10$ cm, and we need to find the perimeter of triangle $PCD$.

Step 2: Using the property of tangents from an external point:
We know that the lengths of two tangents drawn from an external point to a circle are equal. Therefore, we have:
\[ PA = PB = 10 \, \text{cm} \] This means that both tangents from point $P$ are of equal length.

Step 3: Applying the power of a point theorem:
According to the power of a point theorem, if a tangent from an external point intersects a chord at a point, the product of the lengths of the segments formed on the chord is equal to the square of the length of the tangent from the point.
Thus, we can write the following equation for the lengths of the segments on the tangent $PA$: \[ PC \times PD = PA^2 = 10^2 = 100 \, \text{cm}^2 \] This gives us the relation between the lengths of the segments on the tangent at point $E$.

Step 4: Finding the perimeter of $\triangle PCD$:
The perimeter of triangle $PCD$ is the sum of its sides, i.e., $PC + CD + PD$. Since we have the length of the tangents from $P$ to the circle and the power of a point theorem, we can calculate these side lengths.
However, as per the information provided in the question, we need more details about the lengths of the segments $PC$ and $PD$ or additional information about the geometry of the figure to proceed with a numerical calculation.

Conclusion:
The perimeter of triangle $PCD$ depends on the lengths of the segments $PC$ and $PD$. Without additional information, such as the specific location of point $E$ or the radius of the circle, we can conclude that the perimeter involves these lengths. If further data were available, we could compute the exact perimeter using the given relationships.