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 $ riangle PCD$.

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 $	riangle PCD$.

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Step 1: Understanding the given problem: We are given a circle with center $ O $, and two tangents $ PA $ and $ PB $ are drawn from an external point $ P $ to the circle. These tangents meet the circle at points $ A $ and $ B $ respectively. At a point $ E $ on the circle, a tangent is drawn, which intersects $ PA $ at $ C $ and $ PB $ at $ D $. The length of the tangents $ PA $ and $ PB $ is given as $ PA = 10 \, \text{cm} $. We need to find the perimeter of $ \triangle PCD $. Step 2: Use the Power of a Point Theorem: By the Power of a Point Theorem, for any point outside a circle, the product of the lengths of the segments from the point to the points of tangency on each tangent is constant. That is: $$ PC \cdot PA = PD \cdot PB $$ Since $ PA = PB = 10 \, \text{cm} $, we can substitute this into the equation: $$ PC \cdot 10 = PD \cdot 10 $$ Thus, we have: $$ PC = PD $$ This implies that $ C $ and $ D $ are equidistant from the external point $ P $, meaning that $ PC = PD $. Step 3: Perimeter of $ \triangle PCD $: The perimeter of $ \triangle PCD $ is the sum of the lengths of its sides $ PC $, $ CD $, and $ PD $. Since $ PC = PD $, the perimeter is: $$ \text{Perimeter of } \triangle PCD = 2 \times PC + CD $$ To calculate $ CD $, we use the fact that the tangent at point $ E $ intersects $ PA $ and $ PB $, but without additional information such as the specific length of $ CD $, we cannot determine the exact value of $ CD $ directly from the given data. Conclusion: We have found that $ PC = PD $, but we need additional information about the length of $ CD $ to determine the perimeter of $ \triangle PCD $. The perimeter will be $ 2 \times PC + CD $, but the exact value of $ CD $ is not provided in the problem.

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 $\triangle PCD$.

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