To find the length of the diagonal of a rectangle, we can use the Pythagorean theorem in the right-angled triangle formed by the diagonal and the adjacent sides of the rectangle.

Let d represent the length of the diagonal, l represent the length, and w represent the width of the rectangle. Then we have:

l² + w² = d²

Given the length l = 12 cm and width w = 4 cm, we have:

12² + 4² = d²
144 + 16 = d²
160 = d²

Taking the square root of both sides:

d = √160 = 4√10

So, the length of the diagonal is 4√10 cm. The correct answer is B) 4√10.

Since angle A is 90 degrees, triangle ABC is a right-angled triangle with the right angle at vertex A. We can use the Pythagorean Theorem (a² + b² = c²) to find the length of BC, where a and b are the two legs of the right-angled triangle and c is the hypotenuse.

In triangle ABC, a = AB = 5, and b = AC = 12. We have to find the length of the hypotenuse BC.

5² + 12² = c²
25 + 144 = c²
169 = c²

Taking the square root of both sides:

c = 13

So, the length of BC is 13. The correct answer is C) 13.

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

area = (1/2) × base × height

In a right triangle, the base and height are the legs of the triangle. In this case, the legs are 5 cm and 12 cm:

area = (1/2) × 5 × 12
area = (1/2) × 60
area = 30

So, the area of the triangle is 30 square centimeters. The correct answer is B) 30.

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.

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

Area = (1/2) × d₁ × d₂

where d₁ and d₂ are the lengths of the two diagonals.

Given that the diagonals are of lengths 10 and 14, we can calculate the area as follows:

Area = (1/2) × 10 × 14
Area = 5 × 14
Area = 70

So, the area of the rhombus is 70. The correct answer is D) 70.