To find the area of a circle with a given diameter, we first need to find the radius by dividing the diameter by 2:

radius = diameter / 2
radius = 18 / 2
radius = 9

Now, we can use the formula for the area of a circle:

Area = πr²

Substitute r = 9 into the formula:

Area = π(9²)
Area = 81π

So, the area of the circle is 81π. The correct answer is B) 81π.

The area of a parallelogram can be found using the formula:

Area = base × height

Given the side lengths 4 and 6 and the angle of 120 degrees, we can find the height by using trigonometry:

height = shorter_side × sin(angle)

In this case, the shorter side is 4 and the angle is 120 degrees:

height = 4 × sin(120)
height = 4 × (√3 / 2)
height = 2 √3

Now, we can find the area:

Area = 6 × 2 √3
Area = 12 √3

The area of the parallelogram is 12 √3. The correct answer is B) 12 √3.

To find the length of the hypotenuse in a right-angled triangle, we can use the Pythagorean theorem:

a² + b² = c²

where a and b are the lengths of the legs of the triangle, and c is the length of the hypotenuse.

Given the lengths of the legs a = 15 cm and b = 20 cm, we have:

15² + 20² = c²
225 + 400 = c²
625 = c²

Taking the square root of both sides:

c = 25

So, the length of the hypotenuse is 25 cm. The correct answer is B) 25.

Using the Law of Sines in triangle PQR, we have:

sin(P) / PQ = sin(Q) / QR

Given the angle P is 30 degrees, angle Q is 60 degrees, and side PQ has a length of 10, we can substitute the values into the formula:

sin(30) / 10 = sin(60) / QR

Simplifying:

1/2 / 10 = √3/2 / QR

Now, we can cross-multiply to solve for QR:

QR = (10 * √3/2) / (1/2)
QR = 10√3

So, the length of side QR is 10√3. The correct answer is B) 10√3.

When two circles with equal radii intersect, the line segment connecting their centers (AB) is perpendicular to the line segment joining the points of intersection (A and B). This forms a right triangle with hypotenuse AB and legs equal to the radii of the circles.

Since the distance between the centers of the circles is 12, and the radii are equal, we can use the Pythagorean theorem to find the length of AB. Let r be the radius of each circle.

According to the Pythagorean theorem:

(AB)² = (2r)²

Since the distance between the centers is 12, which is twice the radius:

(AB)² = (12)²
(AB)² = 144

Now, take the square root of both sides to find the length of AB:

AB = √144
AB = 12

So the length of AB is 12 units.