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.
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You've sent the same question again, but no worries, I'll provide the solution once more! Here's how to plot the points and prove that ABCD is a rectangle: Step 1: Plot the points on a graph. To plot the points A(3, 2), B(-2, 2), C(-2, -3), and D(3, -3) on a graph paper: • For A(3, 2), start at the origin (0,0), move 3 units to the right along the x-axis, then 2 units up along the y-axis. • For B(-2, 2), start at the origin, move 2 units to the left along the x-axis, then 2 units up along the y-axis. • For C(-2, -3), start at the origin, move 2 units to the left along the x-axis, then 3 units down along the y-axis. • For D(3, -3), start at the origin, move 3 units to the right along the x-axis, then 3 units down along the y-axis. Connect the points in order A B C D A to form the quadrilateral. Step 2: Calculate the lengths of the sides using the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). Length of AB: AB = sqrt((-2 - 3)^2 + (2 - 2)^2) = sqrt((-5)^2 + (0)^2) = sqrt(25) = 5 units Length of BC: BC = sqrt((-2 - (-2))^2 + (-3 - 2)^2) = sqrt((0)^2 + (-5)^2) = sqrt(25) = 5 units Length of CD: CD = sqrt((3 - (-2))^2 + (-3 - (-3))^2) = sqrt((5)^2 + (0)^2) = sqrt(25) = 5 units Length of DA: DA = sqrt((3 - 3)^2 + (2 - (-3))^2) = sqrt((0)^2 + (5)^2) = sqrt(25) = 5 units Since AB = CD = 5 units and BC = DA = 5 units, the opposite sides are equal. In fact, all four sides are equal. Step 3: Calculate the lengths of the diagonals AC and BD. Length of AC: AC = sqrt((-2 - 3)^2 + (-3 - 2)^2) = sqrt((-5)^2 + (-5)^2) = sqrt(25 + 25) = sqrt(50) units Length of BD: BD = sqrt((3 - (-2))^2 + (-3 - 2)^2) = sqrt((5)^2 + (-5)^2) = sqrt(25 + 25) = sqrt(50) units Since AC = BD = sqrt(50) units, the diagonals are equal. Step 4: Conclusion. A quadrilateral is a rectangle if its opposite sides are equal and its diagonals are equal. From Step 2, we found that AB = CD and BC = DA. From Step 3, we found that AC = BD. Therefore, ABCD is a rectangle. (Specifically, since all four sides are equal and the diagonals are equal, ABCD is a square, which is a special type of rectangle). The proof that ABCD is a rectangle is complete. Last free one today — make it count tomorrow, or type /upgrade for unlimited.