Tangent To A Circle

The area enclosed between the square and the circle can be found by subtracting the area of the circle from the area of the square.
1. The side length of the square is 7 cm. Therefore, the area of the square is: 2. The circle is inscribed in the square, so its diameter is equal to the side of the square, i.e., 7 cm. The radius of the circle is half of the diameter: 3. The area of the circle is given by the formula: 4. The area enclosed between the square and the circle is the difference between the area of the square and the area of the circle: Approximating :

The correct option is (D):

In this problem, we are given a circle with a radius of 8 cm, and a tangent drawn from an external point to the circle with a length of 15 cm. The distance between the center of the circle and the external point can be found using the Pythagorean theorem. The radius of the circle, the length of the tangent, and the distance between the center of the circle and the external point form a right triangle, where:
One leg is the radius ( cm),
The other leg is the length of the tangent ( cm),
The hypotenuse is the distance from the center of the circle to the external point . According to the Pythagorean theorem: Substituting the known values:

The correct option is (C):

Correct answer: 1

Explanation:
From a point on the circle, exactly one tangent can be drawn.
This tangent is perpendicular to the radius at the point of contact.

Hence, the number of tangents from a point on the circle is 1.

We can solve this problem using vectors instead of the Pythagorean theorem. The key idea is to consider the vectors in the plane and use the fact that is perpendicular to , making triangle a right triangle.

1. Define vectors:

   Let the vector represent the radius of the circle, which is 3 cm. Let the vector represent the line from point A to P, which is 4 cm. Since is perpendicular to , we know that is orthogonal to .

2. Compute the length of :

   We apply the Pythagorean theorem in terms of vectors. The length of vector is the magnitude of the resultant of the vectors and , which are perpendicular:

   Substituting the known values:

3. Now compute :

   Since is also a radius of the circle and has the same magnitude as , we know:

   The vector is the difference between and , so we have:

Final Answer:

The final answer is cm.

Given: Two concentric circles with a chord of length 24 cm in the larger circle, which acts as a tangent to the smaller circle of radius 5 cm.

Step 1: Understanding the Geometry

Let be the common center of both circles, and let be the chord of the larger circle that becomes a tangent to the smaller circle at some point .

• cm (radius of the smaller circle)
• cm (chord length of the larger circle)

Step 2: Applying the Perpendicular Bisector Theorem

The line is perpendicular to the chord , which means it bisects at point , where:

Let be the radius of the larger circle ( ). Using the right-angled triangle :

Final Answer: 13 cm

In the given figure, and are the tangents to the circle with center .

If , then

Since and are tangents to the circle at points and respectively, we know that and .

Thus, and .

Now consider the quadrilateral . The sum of the angles in a quadrilateral is .

Therefore, Thus, .

Step 1: Understanding the Tangent-Radius Relationship.
At the point where a tangent touches the circle, the radius drawn to the point of contact is always perpendicular to the tangent. This is a well-established property of tangents to a circle.
Step 2: Applying the Perpendicularity Rule.
Since the radius at the point of contact forms a right angle with the tangent, the angle between the tangent and the radius is always .
Step 3: Conclusion.
Therefore, the angle made by the tangent at any point of the circle with the radius at the point of contact is .

Step 1: Understanding the Geometry of Tangents.
When a point lies outside the circle, it can have a maximum of two tangents drawn from it. These tangents touch the circle at two distinct points.
Step 2: Explanation of Tangent Construction.
A tangent is a straight line that touches a circle at exactly one point. The line is perpendicular to the radius at the point of contact. From any point outside the circle, two tangents can be drawn. Both of them will touch the circle at two distinct points, and each will form a right angle with the radius at the point of contact.
Step 3: Conclusion.
Therefore, the number of tangents that can be drawn from a point outside the circle is two.