A field is in the shape of a quadrilateral, with two straight footpaths connecting opposite corners. The footpaths cross at right angles. Given just this information, what can you say about the shape of the field? A) It’s a kite, B) It’s a rectangle, C) None of the other statements, D) It’s a rhombus, and E) It’s a square.
Okay then. Let’s look at the question. We’ve got the shape of a quadrilateral. That means it’s a four-sided polygon. Now we’ve got two straight footpaths connecting opposite corners, and those footpaths cross at right angles. So the diagonals of our quadrilateral cut at ninety degrees to each other.
So it could look something like this. We don’t know the lengths of any of the sides. They could all be the same. They could all be different. We’ve not been given that information. But we do know that the diagonals cross at ninety degrees. So let’s go through these different definitions and see which ones match.
Let’s think about the definition of a square then. A square has four sides. They’re all equal in length. All the interior angles are ninety degrees and the diagonals cross at right angles. So this field could be a square. We can’t prove that it’s not a square. But we can’t prove that it is a square because we don’t know what the side lengths are. Now if it turns out that the square is the only quadrilateral whose diagonals cross at right angles, then we might be back in the game to be able to prove that it is a square.
Now if we look at the definition of a kite, it’s a four-sided shape again. It’s got two pairs of adjacent sides, which are equal in length. The diagonals cross at right angles and one pair of opposite angles are equal. So the fact that in a kite, the diagonals cross at right angles and in a square, the diagonals cross at right angles, means that we can’t decide whether it is a kite or it is a square from just the information that we were given. It could still be either of those. So because the diagonals crossing at right angle doesn’t uniquely identify it as a kite, it’s- we can’t say it definitely is a kite. And because it doesn’t uniquely identify as being a square, we can’t say for definite that it’s a square.
So moving on to the definition of a rhombus, this is a four-sided shape with all sides equal in length and the diagonals cross at ninety degrees. But also the opposite sides are parallel, and the opposite angles are equal. Now that all sounds a bit like a square. And in fact, a square is a special case of a rhombus. A rhombus is basically a squashed square. But again, the only information we’ve got from the question, is that the diagonals are at right angles to each other. That’s not gonna uniquely prove that this is a rhombus. So we can’t say for definite that this shape is a rhombus.
So that leaves us with a rectangle, which is a four-sided shape, with all the interior angles equal to ninety degrees. The opposite sides are parallel and equal in length. Now unless we’re looking at the special case of a rectangle, which is a square, in all other rectangles, these angles here at the centre, where the diagonals meet, are not ninety degrees. And the fact that we know from the question that the diagonals meet at ninety degrees, pretty much rules this out as being a rectangle. We can’t say for definite, it is a rectangle.
And that leaves us with C, “None of the other statements”.
We can’t rule out any of the other shapes, but we can’t prove that they definitely are any of the other shapes, just based on the information that we’ve been given in the question.