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: Properties of tangents
The lengths of tangents from an external point are equal:
$PA = PB = 10 \, \text{cm}, \quad PC = PD.$
Step 2: Find the perimeter
Let $PC = PD = x$ (since tangents from an external point are equal). The perimeter of $\triangle PCD$ is:
$\text{Perimeter} = PC + PD + CD = x + x + CD = 2x + CD.$
By symmetry:
$CD = 2x.$
Substitute:
$\text{Perimeter} = 2x + 2x = 4x.$
{Step 3: Relate $x$ to $PA$
Using the geometry of the figure:
$x = \frac{PA}{2} = \frac{10}{2} = 5 \, \text{cm}.$
Step 4: Calculate the perimeter
$\text{Perimeter} = 4x = 4(5) = 20 \, \text{cm}.$
Correct Answer: $20 \, \text{cm}$.

Answer 2

Step 1: Properties of tangents
The lengths of tangents from an external point are equal:
$PA = PB = 10 \, \text{cm}, \quad PC = PD.$
Step 2: Find the perimeter
Let $PC = PD = x$ (since tangents from an external point are equal). The perimeter of $\triangle PCD$ is:
$\text{Perimeter} = PC + PD + CD = x + x + CD = 2x + CD.$
By symmetry:
$CD = 2x.$
Substitute:
$\text{Perimeter} = 2x + 2x = 4x.$
{Step 3: Relate $x$ to $PA$
Using the geometry of the figure:
$x = \frac{PA}{2} = \frac{10}{2} = 5 \, \text{cm}.$
Step 4: Calculate the perimeter
$\text{Perimeter} = 4x = 4(5) = 20 \, \text{cm}.$
Correct Answer: $20 \, \text{cm}$.