Tangent To A Circle

To solve the problem, we need to find the angle formed between the two tangents drawn from a point outside the circle.

1. Geometry of Tangents from External Point:
When two tangents are drawn from an external point to a circle, the triangle formed (in this case ) is isosceles and angle between the tangents at the circle’s center is given.

2. Use the Property of Triangle:
The triangle formed at the center by joining the tangents' points of contact has the angle .
Now, the angle between the tangents at the external point is the external angle for triangle :

Final Answer:
The value of is .

To solve the problem, we analyze the geometry of the figure shown.

1. Understanding the Figure:
From point , two tangents are drawn to a circle — one touches at point and the other at point .
Given that the length of one tangent cm, and we are asked to find .

2. Property of Tangents from an External Point:
The lengths of tangents drawn from an external point to a circle are always equal.
So,

3. Substituting the Value:

Final Answer:
The value of is .

To solve the problem, we need to find the length of a tangent drawn from a point 15 cm away from the center of a circle with radius 9 cm.

1. Understanding the Geometry:
The situation forms a right-angled triangle with:
- Hypotenuse = distance from the external point to the center = 15 cm
- One leg = radius of the circle = 9 cm
- Other leg = length of the tangent (to be found)

2. Applying the Pythagoras Theorem:
Let be the length of the tangent. Then:

3. Solving for :
cm

Final Answer:
The length of the tangent is .

To solve the problem, we need to find the angle between the tangent and the radius of a circle drawn through the point of contact.

1. Fundamental Concept:
A tangent to a circle is always perpendicular to the radius drawn to the point of contact.
This is a standard geometric property of a circle.

2. Therefore:
The angle between the tangent and the radius at the point of contact is .

Final Answer:
The required angle is .

To solve the problem, we need to find the value of angle when the two tangents and are drawn from a point to a circle with center and .

1. Understanding the Geometry:
In the figure, and are tangents to the circle from point . The tangents from a point outside a circle are equal in length and subtend equal angles at the center.

Given:
Since is a radius and is a tangent,
So triangle is a right triangle with and .

2. Finding :
Using the triangle angle sum property in :

3. as Exterior Angle:
Since is the exterior angle of triangle (formed by the tangents and center),
(Half of central angle formed by equal tangents)

Final Answer:

To solve the problem, we need to find the angle between a tangent to a circle and the radius drawn to the point of contact.

1. Geometrical Property of Circle:
A tangent to a circle is always perpendicular to the radius at the point where the tangent touches the circle.

2. Meaning of Perpendicular:
Perpendicular lines meet at a right angle, which is:

Final Answer:
The angle between the tangent and the radius is .

Step 1: Visualize the Problem Setup
Consider a circle with center O and radius r = 9 cm. A point P is 15 cm away from O. A tangent from P touches the circle at point T. Since a tangent is perpendicular to the radius at the point of contact, angle OTP = 90 degrees, forming a right triangle OTP. In this triangle:
- OT = 9 cm (radius),
- OP = 15 cm (distance from the point to the center),
- PT is the length of the tangent, which we need to find.

Step 2: Use the Pythagorean Theorem
In right triangle OTP, apply the Pythagorean theorem:
OP² = OT² + PT²
Rearrange to solve for PT:
PT² = OP² - OT²
Substitute the values:
- OP = 15 cm, so OP² = 15² = 225,
- OT = 9 cm, so OT² = 9² = 81.
Then:
PT² = 225 - 81 = 144
PT = √144 = 12 cm

Step 3: Interpret the Result
The length of the tangent is 12 cm. Since OP = 15 cm is greater than the radius 9 cm, point P is outside the circle, which is consistent with the problem. The calculated length of 12 cm matches option 3 from the given choices: 15 cm, 13 cm, 11 cm, and 12 cm.

Final Answer: The length of the tangent is 12 cm (Option 3).