A shape has diagonals with slopes that are negative reciprocals and opposites sides with equal lengths and equal slopes must be a square.

1 answer

Yes, you are correct. If a shape has diagonals with slopes that are negative reciprocals, and opposite sides with equal lengths and equal slopes, then the shape must be a square.

Let's consider a shape with these properties. Since opposite sides have equal lengths, we can conclude that the shape is symmetric. Now, let's assume that it is not a square.

If the shape is not a square, it means that the opposite sides are not perpendicular. In other words, the slopes of the opposite sides are not negative reciprocals. But we have already stated that the diagonals have slopes that are negative reciprocals. This means that the diagonals and the opposite sides cannot both exist simultaneously.

Therefore, the assumption that the shape is not a square leads to a contradiction. Hence, the shape must be a square.