Given three distinct quadrilaterals. a square, rectangle. and a rhombus. which quadrilaterals must have perpendicular diagonals? 1) the rhombus. only 2) the rectangle and the square 3) the rhombus and the square 4) the rectangle. the rhombus. and the square 3 2 1 4
The Correct Answer and Explanation is:
Correct Answer:
3) The rhombus and the square
Explanation:
To understand which quadrilaterals have perpendicular diagonals, let us briefly examine the properties of each figure:
- Square
A square has all sides equal and all angles equal to 90 degrees. Its diagonals have two important properties:
- They are equal in length
- They bisect each other at right angles (perpendicular)
Thus, the diagonals of a square are always perpendicular to each other.
- Rectangle
A rectangle has opposite sides equal and all angles equal to 90 degrees. Its diagonals:
- Are equal in length
- Bisect each other
However, the diagonals of a rectangle are not necessarily perpendicular unless the rectangle is also a square. So in a general rectangle that is not a square, the diagonals do not intersect at right angles.
- Rhombus
A rhombus has all sides equal, but the angles are not necessarily 90 degrees. The diagonals of a rhombus:
- Bisect each other
- Are perpendicular to each other
Therefore, the diagonals of a rhombus always intersect at right angles, regardless of the angles of the rhombus.
Summary
- Square: Diagonals are perpendicular
- Rectangle: Diagonals are not necessarily perpendicular
- Rhombus: Diagonals are perpendicular
Thus, only the square and the rhombus must have perpendicular diagonals, which corresponds to option 3) The rhombus and the square.
