Complete the coordinate proof of the theorem.

Given: A B C D is a square. Prove: The diagonals of A B C D are perpendicular. Art: A square A B C D is graphed on a coordinate plane. The horizontal x-axis and vertical y-axis are solid. The vertex labeled as A lies on begin ordered pair 0 comma 0 end ordered pair. The vertex labeled as B lies on begin ordered pair a comma 0 end ordered pair. The vertices C and D are unlabeled. Diagonal A C and B D are drawn by dotted lines.



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The coordinates of square ABCD are A(0, 0), B(a, 0), C(, a), and D(0, ).

The slope of AC¯¯¯¯¯ , when simplified, is equal to .

The slope of BD¯¯¯¯¯, when simplified, is equal to −1.

The product of the slopes is equal to .

Therefore, AC¯¯¯¯¯ is perpendicular to BD¯¯¯¯¯.