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
Step 1: Define the complex numbers representing the points A, B, C, and D. Let z_A = 9+j Let z_B = 4+j13 Let z_C = -8+j8 Let z_D = -3-j4 Step 2: Calculate the complex numbers representing the vectors for each side of the quadrilateral. 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), the quadrilateral ABCD is a rhombus. Step 4: Check for perpendicularity of adjacent sides to prove it's a square. We can check if the angle at vertex B is a right angle. This means checking if vector BA is perpendicular to vector BC. Vector BA = z_A - z_B = (9+j) - (4+j13) = (9-4) + j(1-13) = 5 - j12. Vector BC = -12 - j5. To check for perpendicularity, we examine the ratio BCBA. If this ratio is purely imaginary, the vectors are perpendicular. BCBA = (-12 - j5)/(5 - j12) Multiply the numerator and denominator by the conjugate of the denominator, which is 5+j12: (-12 - j5)/(5 - j12) × (5 + j12)/(5 + j12) = ((-12)(5) + (-12)(j12) + (-j5)(5) + (-j5)(j12))/(5^2 + (-12)^2) = (-60 - j144 - j25 - j^260)/(25 + 144) Since j^2 = -1: = (-60 - j169 + 60)/(169) = (-j169)/(169) = -j Since the ratio BCBA is -j, which is a purely imaginary number, the vectors BA and BC are perpendicular. This means that the angle ABC is 90^. Step 5: Conclusion. A quadrilateral with four equal sides (a rhombus) and at least one right angle is a square. Therefore, ABCD is a square. Last free one today — make it count tomorrow, or type /upgrade for unlimited.