Angle AOP is an acute angle .circle (O, OP ) is passing from A on the circle. Line l touches at A. N is perpendicular on line l from P. If AN =15 PN =9 then radius=-----
To find the radius of the circle, we can use the properties of a right triangle.
Let's label the point where the line l touches the circle as point B.
First, we can observe that triangle ANP is a right triangle since N is perpendicular to line l. Therefore, we can apply the Pythagorean theorem to find the length of NP.
Using the Pythagorean theorem:
NP^2 = AN^2 + AP^2
Given that AN = 15 and PN = 9, we can substitute these values into the equation:
NP^2 = 15^2 + 9^2
NP^2 = 225 + 81
NP^2 = 306
Taking the square root of both sides:
NP = sqrt(306)
Now, let's consider triangle NBO. From the given information, we know that AN is a tangent to the circle. By the tangent-secant theorem, we have:
NO × NB = NP^2
Since NB is the radius of the circle, we can substitute it with r:
NO × r = NP^2
Substituting the value of NP we found earlier:
NO × r = sqrt(306)^2
NO × r = 306
Since we are looking for the radius, we can isolate r by dividing both sides of the equation by NO:
r = 306 / NO
However, we don't have the value of NO yet. To find it, we need to use some properties of angles.
Angle AOP is given to be an acute angle. Since line l is tangent to the circle at point A, we can conclude that angle OAB is a right angle. Therefore, triangle AOB is a right triangle.
Using the trigonometric property tangent (tan):
tan(angle AOP) = NO / OA
We can rewrite this equation as:
NO = OA × tan(angle AOP)
Now, we need to find the value of tan(angle AOP). Since we know the lengths of AN and PN, we can use the tangent function to find the value of this angle.
tan(angle AOP) = PN / AN
Substituting the given values:
tan(angle AOP) = 9 / 15
tan(angle AOP) = 0.6
Now we can substitute this value into the equation for NO:
NO = OA × 0.6
To proceed further, we need additional information about the given problem, specifically the relationship between angles and lengths or any other information to find the value of OA.