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
Hey Ansh, good to see you again. Step 1: Identify the given vertices. The three given vertices of the rectangle ABCD are A(1, 1), B(-3, 1), and C(-3, 3). We need to find the coordinates of the fourth vertex, D(x, y). Step 2: Determine the coordinates of the fourth vertex D. For a rectangle ABCD, opposite sides are parallel and equal in length. This means that the vector AB must be equal to the vector DC. First, calculate AB: AB = x_B - x_A \\ y_B - y_A = -3 - 1 \\ 1 - 1 = -4 \\ 0 Next, express DC in terms of D(x, y): DC = x_C - x_D \\ y_C - y_D = -3 - x \\ 3 - y Equating the components of AB and DC: -3 - x = -4 x = 1 3 - y = 0 y = 3 So, the coordinates of the fourth vertex are D(1, 3). (Note: The hint "Coordinates of D are (3, 3)" is inconsistent with ABCD being a rectangle given the points A, B, and C. We proceed with the calculated coordinates for D.) Step 3: Calculate the lengths of the adjacent sides. We can use the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). Length of side AB: AB = sqrt((-3 - 1)^2 + (1 - 1)^2) = sqrt((-4)^2 + 0^2) = sqrt(16) = 4 units Length of side BC: BC = sqrt((-3 - (-3))^2 + (3 - 1)^2) = sqrt(0^2 + 2^2) = sqrt(4) = 2 units Step 4: Calculate the area of the rectangle. The area of a rectangle is given by the product of its length and width. Area = AB × BC = 4 × 2 = 8 square units The coordinates of the fourth vertex are D(1, 3) and the area of rectangle ABCD is 8 square units. Send me the next one 📸