A tangent `PQ` at a point `P` of a circle of radius `5cm` meets a line through the centre `O` at a point `Q` so that `OQ=13cm`. Find the length of `PQ`.

Options

A. 12 cm
B. 7 cm
C. 10 cm
D. 15 cm

Explanation

To find the length of the tangent \( PQ \) from point \( P \) on the circle to point \( Q \) on the line through the center \( O \), we can use the Pythagorean theorem. Here are the steps to solve the problem:

### Step 1: Identify the given values
- Radius of the circle \( r = OP = 5 \, \text{cm} \)
- Length from the center to point \( Q \) \( OQ = 13 \, \text{cm} \)

### Step 2: Set up the triangle
In triangle \( OPQ \):
- \( OP \) is the radius of the circle (5 cm).
- \( OQ \) is the distance from the center to point \( Q \) (13 cm).
- \( PQ \) is the tangent we want to find.

### Step 3: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ OQ^2 = OP^2 + PQ^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + PQ^2 \]
Calculating the squares:
\[ 169 = 25 + PQ^2 \]

### Step 4: Solve for \( PQ^2 \)
Rearranging the equation:
\[ PQ^2 = 169 - 25 \]
\[ PQ^2 = 144 \]

### Step 5: Find \( PQ \)
Taking the square root of both sides:
\[ PQ = \sqrt{144} = 12 \, \text{cm} \]

Thus, the length of the tangent \( PQ \) is \( 12 \, \text{cm} \).

### Final Answer:
The length of \( PQ \) is \( 12 \, \text{cm} \).

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