Given points A(9, 1), B(4, 13), C(-8, 8), and D(-3, -4) in the complex plane, prove that ABCD is a square.

To prove that ABCD is a square, we need to show that all four sides are equal in length and that one of the interior angles is 90^. Let the complex numbers representing the points be: z_A = 9 + j z_B = 4 + j13 z_C = -8 + j8 z_D = -3 - j4 Step 1: Calculate the length of each side. The length of a segment between two complex numbers z_1 and z_2 is given by |z_2 - z_1|. Length of AB: |AB| = |z_B - z_A| = |(4 + j13) - (9 + j)| = |(4 - 9) + j(13 - 1)| = |-5 + j12| |AB| = sqrt((-5)^2 + (12)^2) = sqrt(25 + 144) = sqrt(169) = 13 Length of BC: |BC| = |z_C - z_B| = |(-8 + j8) - (4 + j13)| = |(-8 - 4) + j(8 - 13)| = |-12 - j5| |BC| = sqrt((-12)^2 + (-5)^2) = sqrt(144 + 25) = sqrt(169) = 13 Length of CD: |CD| = |z_D - z_C| = |(-3 - j4) - (-8 + j8)| = |(-3 - (-8)) + j(-4 - 8)| = |5 - j12| |CD| = sqrt((5)^2 + (-12)^2) = sqrt(25 + 144) = sqrt(169) = 13 Length of DA: |DA| = |z_A - z_D| = |(9 + j) - (-3 - j4)| = |(9 - (-3)) + j(1 - (-4))| = |12 + j5| |DA| = sqrt((12)^2 + (5)^2) = sqrt(144 + 25) = sqrt(169) = 13 Since |AB| = |BC| = |CD| = |DA| = 13, all four sides are equal. This means ABCD is a rhombus. Step 2: Check for a right angle. We can determine the angle between two adjacent sides, for example, AB and BC, by examining the ratio of the complex numbers representing these vectors. The vector AB is represented by z_B - z_A = -5 + j12. The vector BC is represented by z_C - z_B = -12 - j5. Consider the ratio (z_C - z_B)/(z_B - z_A): (z_C - z_B)/(z_B - z_A) = (-12 - j5)/(-5 + j12) Multiply the numerator and denominator by the conjugate of the denominator: (-12 - j5)/(-5 + j12) × (-5 - j12)/(-5 - j12) = ((-12)(-5) + (-12)(-j12) + (-j5)(-5) + (-j5)(-j12))/((-5)^2 + (12)^2) = (60 + j144 + j25 + j^2 60)/(25 + 144) = (60 + j169 - 60)/(169) = (j169)/(169) = j Since the ratio (z_C - z_B)/(z_B - z_A) = j, this means that the vector BC is obtained by rotating the vector AB by 90^ counter-clockwise. Therefore, ABC = 90^. Since all four sides of the quadrilateral ABCD are equal in length and one of its interior angles is 90^, ABCD is a square. ABCD is a square. Send me the next one 📸