Which quadrilaterals have perpendicular diagonals?
A. Rhombus, Rectangle, and Kite B. Rectangle, Square, and Isosceles Trapezoid C. Square, Rhombus, and Kite D. Isosceles Trapezoid, Rhombus, and Square
The Correct Answer and Explanation is :
The correct answer is C. Square, Rhombus, and Kite.
To understand why these quadrilaterals have perpendicular diagonals, we need to examine the properties of each shape.
- Square: A square is a special type of rectangle and rhombus. The diagonals of a square not only bisect each other but are also equal in length and perpendicular. This means that they intersect at right angles (90 degrees), forming four right triangles within the square.
- Rhombus: A rhombus is defined as a quadrilateral with all sides of equal length. Like squares, the diagonals of a rhombus bisect each other at right angles. This property is a consequence of the symmetry and the equal lengths of the sides, leading to the diagonals dividing the rhombus into four congruent right triangles.
- Kite: A kite is a quadrilateral with two pairs of adjacent sides that are equal. The diagonals of a kite are also perpendicular to each other. One of the diagonals bisects the other at a right angle. This property results from the symmetry of the kite; the longer diagonal serves as an axis of symmetry, bisecting the shorter diagonal and creating right angles at the point of intersection.
In contrast, other options listed in the question, such as rectangles and isosceles trapezoids, do not have perpendicular diagonals. For example, in a rectangle, the diagonals are equal in length and bisect each other but do not intersect at right angles. An isosceles trapezoid has one pair of parallel sides and does not necessarily have perpendicular diagonals.
In summary, the square, rhombus, and kite uniquely feature perpendicular diagonals due to their specific geometric properties, making option C the correct answer.