The circle of radius 4 cm and centre O has a tangent PR. If `/_POR = 90^@` and OR = 7 cm and `OP=24 cm`. Calculate the length (in cm.) of PR.

Options

A. 20
B. 24
C. 25
D. 35

Answer

To solve the problem, we will use the Pythagorean theorem, which applies to right-angled triangles. Here are the steps:

### Step 1: Understand the Geometry
We have a circle with center O and radius 4 cm. The tangent line PR touches the circle at point R. We know that angle POR is 90 degrees, OR is 7 cm, and OP is 24 cm.

### Step 2: Identify the Triangle
In triangle POR, we have:
- OP (the distance from point O to point P) = 24 cm (hypotenuse)
- OR (the distance from point O to point R) = 7 cm (one leg)
- PR (the length of the tangent) = ? (the other leg)

### Step 3: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ PR^2 + OR^2 = OP^2 \]

### Step 4: Substitute the Known Values
Substituting the values we have:
\[ PR^2 + 7^2 = 24^2 \]

Calculating the squares:
\[ PR^2 + 49 = 576 \]

### Step 5: Solve for PR^2
Now, isolate PR^2:
\[ PR^2 = 576 - 49 \]
\[ PR^2 = 527 \]

### Step 6: Calculate PR
Now, take the square root of both sides to find PR:
\[ PR = \sqrt{527} \]

### Step 7: Simplify the Square Root
Calculating the square root:
\[ PR \approx 22.9 \text{ cm} \]

### Final Answer
The length of PR is approximately 22.9 cm.

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