Let P be the parallelogram ABDC whose diagonals lie on the lines r1 : x =t+1 y =−t+1 ,t∈R and r2 :x=−2t+1 y =t+2 ,t∈R. Knowing that A = (1,1) and that AB ⊂ r, where r is a straight line parallel to the vector (2, 1), determine the vertices B, C and D of P.

To determine the vertices B, C, and D of the parallelogram P, we need to find the intersection points of the lines r1 and r2, and then use these points to construct the parallelogram. 1. Find the intersection point of r1 and r2: We have the equations for r1 and r2 as follows: r1: x = t + 1, y = -t + 1 r2: x = -2t + 1, y = t + 2 To find the intersection point, we can set the x and y values equal to each other: t + 1 = -2t + 1 (for x) -t + 1 = t + 2 (for y) Solving these equations: For x: t + 1 = -2t + 1 3t = 0 t = 0 For y: -t + 1 = t + 2 2t = -1 t = -1/2 So, the intersection point is (t, t) = (0, 0). 2. Now that we have the intersection point, we can use it to construct the parallelogram. Since we know that A = (1, 1), and AB is parallel to the vector (2, 1), we can calculate B as follows: B = A + (2, 1) = (1, 1) + (2, 1) = (3, 2) 3. To find C and D, we need to use the fact that diagonals of a parallelogram bisect each other. So, the midpoint of AC is also the midpoint of BD. Let's find the midpoint of AC: Midpoint of AC = [(1 + 0) / 2, (1 + 0) / 2] = (1/2, 1/2) 4. Now, we can find C and D as follows: C = 2 * Midpoint of AC - A C = 2 * (1/2, 1/2) - (1, 1) = (1, 1) - (1, 1) = (0, 0) D = 2 * Midpoint of AC - B D = 2 * (1/2, 1/2) - (3, 2) = (1, 1) - (3, 2) = (-2, -1) So, the vertices of the parallelogram P are: A = (1, 1) B = (3, 2) C = (0, 0) D = (-2, -1)