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're on a roll — 1.1 Show that PR = QS Given points P(0, 3), Q(4, 1), R(2, -3), and S(-2, -1). Step 1: Calculate the length of PR using the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). PR = sqrt((2 - 0)^2 + (-3 - 3)^2) PR = sqrt((2)^2 + (-6)^2) PR = sqrt(4 + 36) PR = sqrt(40) Step 2: Calculate the length of QS using the distance formula. QS = sqrt((-2 - 4)^2 + (-1 - 1)^2) QS = sqrt((-6)^2 + (-2)^2) QS = sqrt(36 + 4) QS = sqrt(40) Since PR = sqrt(40) and QS = sqrt(40), then PR = QS. 1.2 Determine the coordinates of N, the midpoint of QS. Step 1: Use the midpoint formula M = ((x_1 + x_2)/(2), (y_1 + y_2)/(2)) for points Q(4, 1) and S(-2, -1). N = ((4 + (-2))/(2), (1 + (-1))/(2)) N = ((2)/(2), (0)/(2)) N = (1, 0) The coordinates of N are (1, 0). 1.3 Prove that PN QS. Step 1: Calculate the gradient of PN using points P(0, 3) and N(1, 0). m_PN = (0 - 3)/(1 - 0) = (-3)/(1) = -3 Step 2: Calculate the gradient of QS using points Q(4, 1) and S(-2, -1). m_QS = (-1 - 1)/(-2 - 4) = (-2)/(-6) = (1)/(3) Step 3: Multiply the gradients. m_PN × m_QS = -3 × (1)/(3) = -1 Since the product of their gradients is -1, PN QS. 1.4 Prove that P, N and R are collinear. Step 1: Calculate the gradient of PN using points P(0, 3) and N(1, 0). m_PN = (0 - 3)/(1 - 0) = -3 Step 2: Calculate the gradient of NR using points N(1, 0) and R(2, -3). m_NR = (-3 - 0)/(2 - 1) = (-3)/(1) = -3 Since m_PN = m_NR and they share a common point N, P, N, and R are collinear. 1.5 What kind of quadrilateral is PQRS? Give a reason. Step 1: Determine properties of the diagonals. From 1.1, PR = QS = sqrt(40), so the diagonals are equal in length. From 1.2, N is the midpoint of QS. From 1.4, N is also the midpoint of PR (as M_PR = ((0+2)/(2), (3+(-3))/(2)) = (1,0), which is N). Since the diagonals bisect each other, PQRS is a parallelogram. From 1.3, PN QS. Since N lies on PR, this means the diagonals PR and QS are perpendicular. Step 2: Classify the quadrilateral based on these properties. A parallelogram with equal diagonals is a rectangle. A parallelogram with perpendicular diagonals is a rhombus. A quadrilateral that is both a rectangle and a rhombus is a square. Therefore, PQRS is a square. Reason: The diagonals PR and QS are equal in length, bisect each other, and are perpendicular. Drop the next question!