Diagonals of a rectangle:

The diagonals of a rectangle are the line segments that connect opposite corners of the rectangle. In a rectangle, there are two diagonals: one connecting the top left corner to the bottom right corner, and another connecting the top right corner to the bottom left corner.

Properties of diagonals in a rectangle:

1. Equal in length: The diagonals of a rectangle are always equal in length. This means that the length of the diagonal connecting the top left corner to the bottom right corner is the same as the length of the diagonal connecting the top right corner to the bottom left corner.

2. Form right angles: The diagonals of a rectangle form right angles with each other. This means that the angle between the diagonal connecting the top left corner to the bottom right corner and the diagonal connecting the top right corner to the bottom left corner is always 90 degrees.

Explanation of the correct answer:

The correct answer is option 'A', which states that the diagonals of a rectangle are equal to each other. This is a fundamental property of rectangles, and it can be proven mathematically.

Proof:
Let's consider a rectangle ABCD, where AB and CD are the parallel sides and AD and BC are the other two sides. The diagonals of this rectangle are AC and BD.

To prove that AC is equal to BD, we can use the Pythagorean theorem. According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In rectangle ABCD, AC and BD are both diagonals. Therefore, we can consider triangles ACD and BCD, where AC and BD are the hypotenuses. Since both triangles share the side CD, the lengths of AD and BC are the same in both triangles.

Using the Pythagorean theorem in triangles ACD and BCD:
- In triangle ACD: AC^2 = AD^2 + CD^2
- In triangle BCD: BD^2 = BC^2 + CD^2

Since AD = BC (as they are opposite sides of a rectangle), the equations can be rewritten as:
- AC^2 = AD^2 + CD^2
- BD^2 = AD^2 + CD^2

By comparing these equations, we can see that AC^2 = BD^2, which implies that AC = BD. Therefore, the diagonals AC and BD of a rectangle are equal in length.

Conclusion:

The correct answer is option 'A', which states that the diagonals of a rectangle are equal to each other. This property can be proven using the Pythagorean theorem and is a fundamental characteristic of rectangles.