This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To prove that ABCD is a square, we need to show that all four sides have equal length and that adjacent sides are perpendicular. Step 1: Define the complex numbers for points A, B, C, D. Let the complex numbers be: z_A = 9+j z_B = 4+j13 z_C = -8+j8 z_D = -3-j4 Step 2: Calculate the complex numbers representing the vectors for each side. Vector AB = z_B - z_A = (4+j13) - (9+j) = (4-9) + j(13-1) = -5 + j12 Vector BC = z_C - z_B = (-8+j8) - (4+j13) = (-8-4) + j(8-13) = -12 - j5 Vector CD = z_D - z_C = (-3-j4) - (-8+j8) = (-3-(-8)) + j(-4-8) = 5 - j12 Vector DA = z_A - z_D = (9+j) - (-3-j4) = (9-(-3)) + j(1-(-4)) = 12 + j5 Step 3: Calculate the magnitude (length) of each side. The magnitude of a complex number x+jy is sqrt(x^2+y^2). Length of AB: |AB| = |-5 + j12| = sqrt((-5)^2 + (12)^2) = sqrt(25 + 144) = sqrt(169) = 13 Length of BC: |BC| = |-12 - j5| = sqrt((-12)^2 + (-5)^2) = sqrt(144 + 25) = sqrt(169) = 13 Length of CD: |CD| = |5 - j12| = sqrt((5)^2 + (-12)^2) = sqrt(25 + 144) = sqrt(169) = 13 Length of DA: |DA| = |12 + j5| = sqrt((12)^2 + (5)^2) = sqrt(144 + 25) = sqrt(169) = 13 Since all four sides have equal length (13 units), ABCD is either a rhombus or a square. Step 4: Check for perpendicularity of adjacent sides. Two vectors (x_1, y_1) and (x_2, y_2) are perpendicular if their dot product x_1 x_2 + y_1 y_2 = 0. Let's check vectors AB = -5 + j12 (corresponding to (-5, 12)) and BC = -12 - j5 (corresponding to (-12, -5)). Dot product of AB and BC: (-5)(-12) + (12)(-5) = 60 - 60 = 0 Since the dot product is 0, AB is perpendicular to BC. This means that angle ABC is 90^. Step 5: Conclude that ABCD is a square. Since all four sides of the quadrilateral ABCD are equal in length and at least one interior angle is 90^, the quadrilateral ABCD is a square. That's 2 down. 3 left today — send the next one.