Circles

Step 1: Determine the side length of the square.

Since the circles touch each other, the distance between the centers of two adjacent circles is equal to twice the radius of the circles. Thus, the side length of the square is:

Step 2: Calculate the area of the square.

The area of a square is given by:

Step 3: Calculate the total area of the four quadrants of the circles.

Each circle has a radius of , so the area of one circle is:

Using :

Each quadrant is one-fourth of a circle, and there are four quadrants (one from each circle). Thus, the total area of the four quadrants is:

Step 4: Find the area of the shaded region.

The shaded region is the area of the square minus the total area of the four quadrants:

Substitute the values:

Final Answer: The area of the shaded region is , which corresponds to option .

Step 1: Recall the formula for the circumference of a circle.

The circumference of a circle is given by:

where is the radius of the circle. Substituting :

Solve for :

Using :

Step 2: Find the area of the circle.

The area of a circle is given by:

Substitute :

Using :

Step 3: Find the area of a quadrant.

A quadrant is one-fourth of the circle's area:

Final Answer: The area of the quadrant is , which corresponds to option .

When two circles touch internally, it means that one circle lies completely inside the other, and they share exactly one point of contact. In this case:

• The distance between the centers of the two circles is equal to the difference of their radii.
• No external tangents can be drawn because one circle is entirely inside the other.
• Only one direct common tangent exists, which passes through the single point of contact between the two circles.

For two circles that touch internally, there is exactly one common tangent, which is the tangent at their point of contact.

Final Answer: The number of common tangents is .

Step 1: Recall the formula for the length of a tangent.

The length of a tangent from a point outside a circle to the point of tangency is given by:

where:

• is the distance from the external point to the center of the circle, and
• is the radius of the circle.

Step 2: Substitute the given values.

Here, and . Substituting these into the formula:

Step 3: Simplify the calculation.

Final Answer: The length of the tangent is .

We are given a circle with radius 8 cm and a point that is 17 cm away from the center of the circle. We need to find the length of the tangent drawn from point to the circle. We can use the Pythagorean theorem to solve this. In the right triangle formed by the radius of the circle, the line joining the center to the point , and the tangent from point to the circle, we know:
The distance from to is 17 cm.
The radius of the circle (distance from to ) is 8 cm.
The length of the tangent (from to ) is what we need to find. Let the length of the tangent be . According to the Pythagorean theorem:

The correct option is (C):

Given: A square of side 10 cm.

Step 1: Determine the Radius of the Inscribed Circle

The largest possible circle that can be inscribed in the square will have its diameter equal to the side of the square.

Step 2: Calculate the Area of the Circle

Final Answer: cm²

Given: A chord of a circle with radius cm subtends an angle at the center.

Step 1: Use chord length formula

where cm and .

Step 2: Substitute values

Final Answer: 6 cm

Step 1: Use the distance formula between each pair of points.
The distance formula between two points and is: Let the points be: Compute distances:

Step 2: Check for collinearity.
If the points were collinear, the sum of the lengths of the two shorter sides would equal the third.
So the points are not collinear.

Step 3: Check for square or parallelogram.
For a square or parallelogram, there must be four points. Here, we only have three points.
So, they must form a triangle.

Step 1: Formula for the Length of an Arc. The length of an arc of a sector is given by the formula: where is the central angle, and is the radius of the circle.
Step 2: Substituting the Given Values. Here, the radius and the angle . Substituting these values into the formula, we get:
Step 3: Conclusion. Thus, the length of the arc is .