Given points P(1,1), Q(0,-2), and the equations of lines PR (y = -x + 2) and QR (x + 3y + 6 = 0), find the gradient of PQ, prove that angle PQR is 90 degrees, and determine the coordinates of R.

Step 1: Identify the coordinates of P and Q. P is (1, 1) and Q is (0, -2). Step 2: Apply the gradient formula m = (y_2 - y_1)/(x_2 - x_1). m_QP = (1 - (-2))/(1 - 0) m_QP = (1 + 2)/(1) m_QP = (3)/(1) m_QP = 3 The gradient of QP is 3. 6.2 Prove that PQR = 90^. Step 1: Determine the gradient of PQ. From 6.1, m_PQ = 3. Step 2: Determine the gradient of QR. The equation of line QR is x + 3y + 6 = 0. Rearrange the equation into the form y = mx + c: 3y = -x - 6 y = -(1)/(3)x - 2 The gradient of QR is m_QR = -(1)/(3). Step 3: Calculate the product of the gradients of PQ and QR. m_PQ × m_QR = 3 × (-(1)/(3)) m_PQ × m_QR = -1 Since the product of the gradients of PQ and QR is -1, the lines PQ and QR are perpendicular. Therefore, PQR = 90^. 6.3 Determine the coordinates of R. Step 1: Identify the equations of lines PR and QR. Equation of PR: y = -x + 2 (1) Equation of QR: x + 3y + 6 = 0 (2) R is the intersection point of these two lines. Step 2: Substitute equation (1) into equation (2). Substitute y = -x + 2 into x + 3y + 6 = 0: x + 3(-x + 2) + 6 = 0 x - 3x + 6 + 6 = 0 -2x + 12 = 0 -2x = -12 x = (-12)/(-2) x = 6 Step 3: Substitute the value of x back into equation (1) to find y. y = -x + 2 y = -(6) + 2 y = -6 + 2 y = -4 The coordinates of R are (6, -4) . 6.4 Calculate the length of PR. Leave your answer in surd form. Step 1: Identify the coordinates of P and R. P is (1, 1) and R is (6, -4). Step 2: Apply the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). PR = sqrt((6 - 1)^2 + (-4 - 1)^2) PR = sqrt((5)^2 + (-5)^2) PR = sqrt(25 + 25) PR = sqrt(50) Step 3: Simplify the surd. PR = sqrt(25 × 2) PR = 5sqrt(2) The length of PR is 5sqrt(2) units. 6.5 Determine the equation of the line (s) in the form y = mx + c. (s) is perpendicular to line y = -x + 2 and passes through the point P. Step 1: Determine the gradient of line (s). The given line is y = -x + 2, so its gradient is m_1 = -1. Line (s) is perpendicular to this line. If m_s is the gradient of line (s), then m_s × m_1 = -1. m_s × (-1) = -1 m_s = 1 Step 2: Use the point-slope form y - y_1 = m(x - x_1) with point P(1, 1) and gradient m_s = 1. y - 1 = 1(x - 1) y - 1 = x - 1 y = x - 1 + 1 y = x The equation of line (s) is y = x. 6.6 Calculate the size of . Step 1: Identify the gradients of the lines forming angle = PRQ. The angle is formed by lines PR and QR. From the equation of PR, y = -x + 2, the gradient is m_PR = -1. From the equation of QR, x + 3y + 6 = 0 y = -(1)/(3)x - 2, the gradient is m_QR = -(1)/(3). Step 2: Use the formula for the angle between two lines, = | (m_1 - m_2)/(1 + m_1 m_2) |. Let