PR is tangent to a circle, with centre O and radius 4 cm, at point Q. If `/_POR = 90^@, OR = 5 cm `and `OP = 20/3 cm`, then (in cm) the length of PR :

Options

A. 3
B. `16/3`
C. `23/3`
D. `25/3`

Answer

To solve the problem, we will follow these steps:

1. **Understand the Geometry**: We have a circle with center O and radius 4 cm. The line PR is tangent to the circle at point Q. We know that the radius at the point of tangency is perpendicular to the tangent line. Therefore, angle OQR = 90°.

2. **Identify Given Values**:
- Radius (OQ) = 4 cm
- Length OR = 5 cm
- Length OP = 20/3 cm

3. **Use Pythagorean Theorem**: In triangle OPR, we can apply the Pythagorean theorem since it is a right triangle (angle OQR = 90°). The sides are:
- OR (one leg) = 5 cm
- OP (hypotenuse) = 20/3 cm
- PR (the other leg, which we need to find)

According to the Pythagorean theorem:
\[ OP^2 = OR^2 + PR^2 \]

4. **Substitute the Values**:
\[ \left(\frac{20}{3}\right)^2 = 5^2 + PR^2 \]
\[ \frac{400}{9} = 25 + PR^2 \]

5. **Convert 25 to a Fraction**:
\[ 25 = \frac{225}{9} \]
So, we rewrite the equation:
\[ \frac{400}{9} = \frac{225}{9} + PR^2 \]

6. **Isolate PR^2**:
\[ PR^2 = \frac{400}{9} - \frac{225}{9} \]
\[ PR^2 = \frac{175}{9} \]

7. **Take the Square Root**:
\[ PR = \sqrt{\frac{175}{9}} = \frac{\sqrt{175}}{3} \]
Simplifying \(\sqrt{175}\):
\[ \sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7} \]
Therefore:
\[ PR = \frac{5\sqrt{7}}{3} \]

8. **Final Answer**:
The length of PR is \(\frac{5\sqrt{7}}{3}\) cm.