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in Mathematics by (53.3k points)

If A(3, 4) and C(1, −1) are the ends of a diagonal of a square ABCD, then find the equations of the sides of the square.

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See Fig.Equation of the line AC is

y - 4 = 4 + 1/3 - 1(x - 3)

2y - 8 = 5x - 15

5x - 2y - 7 = 0

 Its slope is 5/2. Let the slope of side CD be m. Then

 Therefore, equations of CB and CD are

 Similarly, the equations of AB and BD, respectively, are

 Therefore, the equations of the sides are 3x − 7y − 10 = 0, 

7x + 3y − 4 = 0, 3x − 7y − 19 = 0 and 7x + 3y − 33 = 0

 Aliter (Using Complex Numbers): Let A = 3 + 4i, C = 1 − i, M = midpoint of AC = 2 + 3/2i. Let B denote Z. Rotate (bar)MA  about M through 90° in the anticlockwise sense so that

 Now, we can write the sides equations because the vertices are obtained as A(3, 4), B(−1/2, 5/2), C(1, −1), D(9/2, 1/2).

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