Final answer:
To prove that the diagonals of a square are perpendicular, we need to show that the slopes of the diagonals are negative reciprocals of each other.
Step-by-step explanation:
To prove that the diagonals of a square are perpendicular, we need to show that the slopes of the diagonals are negative reciprocals of each other.
Let's consider a square ABCD with coordinates A(a, a), B(-a, a), C(-a, -a), and D(a, -a).
The slope of diagonal AC is (a - (-a))/(a - a), which simplifies to 2a/0, which is undefined. This means the diagonal AC is a vertical line, and its slope does not exist.
The slope of diagonal BD is (a - (-a))/(-a - (-a)), which simplifies to 2a/(-2a), which is -1.
Since the slope of AC does not exist and the slope of BD is -1, we can conclude that the diagonals AC and BD are perpendicular to each other.