Which of the following quadrilaterals has diagonals that bisect each other, perpendicular and congruent
The Correct Answer and Explanation is:
Correct Answer: Square
A square is the quadrilateral that has diagonals that bisect each other, are perpendicular, and are congruent. Let’s explore why.
First, we need to understand the meaning of each property:
- Bisect each other: This means each diagonal cuts the other into two equal halves.
- Perpendicular: This means the diagonals intersect at a right angle (90 degrees).
- Congruent: This means both diagonals are of the same length.
Let’s consider common types of quadrilaterals:
- Parallelogram: Diagonals bisect each other but are not necessarily perpendicular or congruent.
- Rectangle: Diagonals are congruent and bisect each other but do not intersect at a right angle.
- Rhombus: Diagonals bisect each other and are perpendicular, but they are not necessarily congruent.
- Square: Diagonals bisect each other, are perpendicular, and are congruent.
Only the square satisfies all three conditions at once.
A square is a special case of both a rectangle and a rhombus. It inherits the properties of a rectangle, which has equal diagonals, and of a rhombus, which has perpendicular diagonals that bisect each other. Because all sides of a square are equal and all angles are right angles, its diagonals have equal length and meet at 90 degrees. Also, the symmetry in a square ensures the diagonals split each other into two equal parts at the center.
This combination of properties is unique to the square. No other quadrilateral meets all three conditions simultaneously. The geometric precision of a square makes it an ideal figure when all these diagonal properties are required. Therefore, among the given choices, the square is the correct answer.
