Question

A quadrilateral is a ______ if and only if its diagonals are perpendicular?

Answer 1

Orthodiagonal quadrilateral (sometimes known as a perpendicular quad)

Explanation:

My immediate thought was 'kite', but orthodiagonal quadrilaterals cover more cases, e.g.:

Any orthodiagonal quadrilateral can be rotated and translated so that one of its diagonals lies along the `x` axis and the other along the `y` axis, intersecting at the origin. Then the vertices are at `(a, 0)`, `(0, b)`, `(-c, 0)` and `(0, -d)` for some non-zero numbers `a`, `b`, `c` and `d`.

If the orthodiagonal quadrilateral is convex then this can be done with `a, b, c, d > 0`.

A bilaterally symmetric convex orthodiagonal quadrilateral is a kite. Any kite can be translated and rotated so that its vertices are `(a, 0)`, `(0, b)`, `(-c, 0)` and `(0, -d)`, where `a, b, c, d > 0` and `a = c`.

If `a=c` and `b=d` then the quadrilateral is a rhombus.

If `a=b=c=d` then the quadrilateral is a square.

Answer 2

Rhombus, Square

Answer 3

There is not enough information to decide which single quadrilateral this could be.

Explanation:

It could be an irregular quadrilateral whose diagonals happen to intersect at 90°.

Similarly a trapezium, concave quadrilateral, kite, rhombus or square.

It is not given whether the diagonals are equal, or bisect each other, are line of symmetry of the quadrilateral in questions, nor whether the quad itself has equal or parallel sides.

There are therefore multiple possibilities.