a There are several properties that guarantee that a quadrilateral is a square.
Working with Sides
We need to do several checks until we can be sure that the quadrilateral we have is a square.
Using the Distance Formula, we can check the lengths of the sides. If all four lengths are the same, then according to Theorem 6-8 it is a parallelogram, and by definition, it is a rhombus.
If a parallelogram has four congruent sides and four right angles, then by definition it is a square.
Note that checking one right angle is enough, so no need to find all four slopes.
Working with Diagonals
If we work with the diagonals, we still need several steps.
Using the Midpoint Formula, we can check the midpoints of the diagonals. If the midpoints are the same, then the diagonals bisect each other. According to Theorem 6-11, this means that the quadrilateral is a parallelogram.
Using the Distance Formula, we can check the lengths of the diagonals. If the two lengths are the same, then according to Theorem 6-18 the parallelogram is a rectangle.
Using the Slope Formula, we can check the slopes of the diagonals. If the slopes are negative reciprocals, then the diagonals are perpendicular, so according to Theorem 6-16 the parallelogram is a rhombus.
If a parallelogram is a rectangle and a rhombus, then it is a square.
b Both methods involve several steps. In both cases we needed to check that the quadrilateral is a parallelogram, a rhombus, and a rectangle. Let's summarize the steps needed.
Method
Distance Formula
Slope Formula
Midpoint Formula
Working with Sides
4 times
2 times
not used
Working with Diagonals
2 times
2 times
2 times
We can see that there is no big difference in the number of calculations needed.
If we prefer using the Midpoint Formula over the Distance Formula, then working with the diagonals is slightly more efficient.
If we prefer using two formulas instead of three, then working with sides might be considered more efficient.
