Given the vertices of a triangle P(1, -3), Q(7, 5), and R(-3, -5).

Here's the solution to the problem: Given the vertices of a triangle P(1, -3), Q(7, 5), and R(-3, -5). i) Express PQ and PR as column vectors. Step 1: Find vector PQ. To find the vector from point P to point Q, subtract the coordinates of P from the coordinates of Q. PQ = Q - P = 7 \\ 5 - 1 \\ -3 = 7-1 \\ 5-(-3) = 6 \\ 5+3 PQ = 6 \\ 8 Step 2: Find vector PR. To find the vector from point P to point R, subtract the coordinates of P from the coordinates of R. PR = R - P = -3 \\ -5 - 1 \\ -3 = -3-1 \\ -5-(-3) = -4 \\ -5+3 PR = -4 \\ -2 ii) Show that triangle PQR is isosceles. To show that a triangle is isosceles, we need to calculate the lengths of its three sides. If at least two sides have equal length, the triangle is isosceles. The length (magnitude) of a vector x \\ y is given by sqrt(x^2 + y^2). Step 1: Calculate the length of side PQ. Using PQ = 6 \\ 8 : |PQ| = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 The length of PQ is 10 units. Step 2: Calculate the length of side PR. Using PR = -4 \\ -2 : |PR| = sqrt((-4)^2 + (-2)^2) = sqrt(16 + 4) = sqrt(20) The length of PR is sqrt(20) units, which can be simplified to 2sqrt(5) units. Step 3: Calculate the length of side QR. First, find the vector QR. Subtract the coordinates of Q from the coordinates of R. QR = R - Q = -3 \\ -5 - 7 \\ 5 = -3-7 \\ -5-5 = -10 \\ -10 Now, calculate the length of QR: |QR| = sqrt((-10)^2 + (-10)^2) = sqrt(100 + 100) = sqrt(200) The length of QR is sqrt(200) units, which can be simplified to 10sqrt(2) units. Step 4: Compare the lengths of the