A shape has diagonals with slopes that are negative reciprocals and opposites sides with equal lengths and equal slopes must be a square.
Explain
This statement is essentially describing the unique properties of a square.
In a square, all four sides are equal in length and have equal slopes. This means that if we take any two adjacent sides and find the slope between them, it will be the same as any other pair of adjacent sides.
The statement also mentions that the diagonals have slopes that are negative reciprocals of each other. This means that if we find the slope of one diagonal and then find the negative reciprocal of that slope, it will be the slope of the other diagonal.
Using these properties, we can conclude that if a shape has sides with equal lengths and equal slopes, and diagonals with slopes that are negative reciprocals of each other, then it must be a square.
This is because no other shape satisfies all these conditions. Other shapes may have equal length sides, but their diagonals will not necessarily have negative reciprocal slopes. Similarly, there may be shapes with diagonals that have negative reciprocal slopes, but their sides will not have equal lengths and equal slopes.
Thus, a shape satisfying all the given conditions can only be a square.