Plot the vertices of triangle PQR.

Here's the solution to the geometry problem. a) Draw PQR on the grid provided. Step 1: Plot the vertices of triangle PQR. The vertices are P(3,2), Q(-1,1), and R(-3,-1). Plot these points on the grid. Step 2: Draw the triangle. Connect the points P, Q, and R with straight lines to form triangle PQR. b) Under a rotation the vertices of P'Q'R' are P'(1,4), Q'(2,0) and R'(4,-1). Find the centre and angle of rotation using points P and Q. Step 1: Find the equation of the perpendicular bisector of PP'. Given P(3,2) and P'(1,4). Midpoint of PP': M_PP' = ((3+1)/(2), (2+4)/(2)) = ((4)/(2), (6)/(2)) = (2,3). Gradient of PP': m_PP' = (4-2)/(1-3) = (2)/(-2) = -1. The gradient of the perpendicular bisector is m_ PP' = -(1)/(m_PP') = -(1)/(-1) = 1. Equation of the perpendicular bisector (using point-slope form y - y_1 = m(x - x_1)): y - 3 = 1(x - 2) y = x - 2 + 3 y = x + 1 (*) Step 2: Find the equation of the perpendicular bisector of QQ'. Given Q(-1,1) and Q'(2,0). Midpoint of QQ': M_QQ' = ((-1+2)/(2), (1+0)/(2)) = ((1)/(2), (1)/(2)). Gradient of QQ': m_QQ' = (0-1)/(2-(-1)) = (-1)/(3). The gradient of the perpendicular bisector is m_ QQ' = -(1)/(m_QQ') = -(1)/(-1/3) = 3. Equation of the perpendicular bisector: y - (1)/(2) = 3(x - (1)/(2)) y - (1)/(2) = 3x - (3)/(2) y = 3x - (3)/(2) + (1)/(2) y = 3x - 1 (**) Step 3: Find the centre of rotation by solving the system of equations. Equate () and (*): x + 1 = 3x - 1 1 + 1 = 3x - x 2 = 2x x = 1 Substitute x=1 into y = x + 1: y = 1 + 1 y = 2 The centre of rotation is (1,2). Step 4: Determine the angle of rotation. Consider the vector from the centre C(1,2) to P(3,2): CP = (3-1, 2-2) = (2,0). Consider the vector from the centre C(1,2) to P'(1,4): CP' = (1-1, 4-2) = (0,2). A rotation that transforms (2,0) to (0,2) is a 90^ counter-clockwise rotation. This can be verified using the rotation matrix for 90^ counter-clockwise: 0 & -1 \\ 1 & 0 2 \\ 0 = 0 \\ 2 . The angle of rotation is 90^ counter-clockwise. c) Triangle PQR is enlarged with scale factor 3 centre (0,0) to give triangle P²Q²R². Draw triangle P²Q²R² and state its co-ordinates. Step 1: Calculate the coordinates of P²Q²R². For an enlargement with scale factor k and centre (0,0), a point (x,y) maps to (kx, ky). P(3,2) P²(3 × 3, 2 × 3) = (9,6) Q(-1,1) Q²(-1 × 3, 1 × 3) = (-3,3) R(-3,-1) R²(-3 × 3, -1 × 3) = (-9,-3) The coordinates are P^2(9,6), Q^2(-3,3), R^2(-9,-3). Step 2: Draw triangle P²Q²R². Plot the points P²(9,6), Q²(-3,3), and R²(-9,-3) on the grid. Connect these points with straight lines to form triangle P²Q²R². d) Triangle P¹Q¹R¹ undergoes reflection in line y = -x to give triangle P³Q³R³. Draw P³Q³R³ and state its coordinates. Step 1: Calculate the coordinates of P³Q³R³. For a reflection in the line y = -x, a point (x,y) maps to (-y, -x). P'(1,4) P³(-4, -1) Q'(2,0) Q³(-0, -2) = (0, -2) R'(4,-1) R³(-(-1), -4) = (1, -4) The coordinates are P^3(-4,-1), Q^3(0,-2), R^3(1,-4). Step 2: Draw triangle P³Q³R³. Plot the points P³(-4,-1), Q³(0,-2), and R³(1,-4) on the grid. Connect these points with straight lines to form triangle P³Q³R³. 3 done, 2 left today. You're making progress.