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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Hey ThEe, good to see you again. Here are the solutions to the questions: Given points: P(0, 3) Q(4, 1) R(2, -3) S(-2, -1) 1.1 Show that PR = QS Step 1: Calculate the length of PR using the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). For PR, use P(0, 3) and R(2, -3): 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. For QS, use Q(4, 1) and S(-2, -1): 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), we have PR = QS. PR = QS = sqrt(40) 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 N, the midpoint of QS, use 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 the gradient formula m = (y_2 - y_1)/(x_2 - x_1). For PN, use P(0, 3) and N(1, 0): m_PN = (0 - 3)/(1 - 0) = (-3)/(1) = -3 Step 2: Calculate the gradient of QS. For QS, use Q(4, 1) and S(-2, -1): m_QS = (-1 - 1)/(-2 - 4) = (-2)/(-6) = (1)/(3) Step 3: Check if the product of the gradients is -1. m_PN × m_QS = -3 × (1)/(3) = -1 Since the product of their gradients is -1, PN is perpendicular to QS. PN QS 1.4 Prove that P, N and R are collinear. Step 1: We already have the gradient of PN from 1.3: m_PN = -3. Step 2: Calculate the gradient of NR. For NR, use N(1, 0) and R(2, -3): m_NR = (-3 - 0)/(2 - 1) = (-3)/(1) = -3 Since m_PN = m_NR = -3 and N is a common point, P, N, and R are collinear. P, N, R are collinear 1.5 What kind of quadrilateral is PQRS? Give a reason. Step 1: Determine if the diagonals bisect each other. From 1.2, N is the midpoint of QS, and N = (1, 0). Let's find the midpoint of PR using P(0, 3) and R(2, -3): M_PR = ((0 + 2)/(2), (3 + (-3))/(2)) = ((2)/(2), (0)/(2)) = (1, 0) Since the midpoint of PR is also (1, 0), N is the common midpoint of both diagonals PR and QS. This means the diagonals bisect each other, so PQRS is a parallelogram. Step 2: Consider the lengths of the diagonals. From 1.1, we showed that PR = QS = sqrt(40). A parallelogram with equal diagonals is a rectangle. Step 3: Consider the relationship between the diagonals. From 1.3, we showed that m_PN × m_QS = -1. Since N lies on PR (from 1.4), this implies that the diagonal PR is perpendicular to the diagonal QS. A parallelogram with perpendicular diagonals is a rhombus. Step 4: Conclude the type of quadrilateral. Since PQRS is both a rectangle (diagonals are equal) and a rhombus (diagonals are perpendicular), it must be a square. The quadrilateral PQRS is a square. Reason: Its diagonals bisect each other (making it a parallelogram), are equal in length (making it a rectangle), and are perpendicular (making it a rhombus). A quadrilateral with all these properties is a square. Drop the next question 📸