Angle AOP is an acute angle .circle (O, OP ) is passing from A. Line l touches at A. No is perpendicular on line l from P. If AN =15 PN =9 then radius=-----

I can't figure out what

circle (O, OP ) is passing from A

means.

To find the radius of the circle (O, OP), we can use the properties of tangents and perpendiculars in circles.

Let's use the given information to find the solution step by step:

1. First, we know that when a tangent line intersects a radius of a circle, the angle between the tangent line and the radius is 90 degrees. Therefore, angle AON is a right angle.

2. Since AN is perpendicular to line l, angle ANP is also a right angle.

3. Angle AOP is an acute angle, which means it measures less than 90 degrees. Therefore, angle PON must measure more than 90 degrees.

4. From the given information, we know that AN = 15 and PN = 9.

5. Notice that triangle AON forms a right-angled triangle with sides AN and ON. Using the Pythagorean theorem, we can find the length of ON:

ON² = AO² - AN²
ON² = r² - 15²
ON² = r² - 225

6. Similarly, triangle PON forms a right-angled triangle with sides PN and ON. Using the Pythagorean theorem, we can find the length of ON:

ON² = OP² - PN²
ON² = r² - 9²
ON² = r² - 81

7. Since both expressions for ON² are equal, we can equate them and solve for r:

r² - 225 = r² - 81
Subtracting r² from both sides:
-225 = -81
This statement is not true.

By examining the equations, we can see that there is an error.

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