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 anujt1724 — ready when you are. Here are the solutions for questions 35 and 36. Question 35: Rhombus a) Find the area of a rhombus if its vertices are A(3, 0), B(4, 5), C(-1, 4) and D(-2, -1). Step 1: Calculate the length of the diagonals. The diagonals of the rhombus are AC and BD. Length of diagonal AC: AC = sqrt((-1-3)^2 + (4-0)^2) AC = sqrt((-4)^2 + (4)^2) AC = sqrt(16 + 16) AC = sqrt(32) = 4sqrt(2) units Length of diagonal BD: BD = sqrt((-2-4)^2 + (-1-5)^2) BD = sqrt((-6)^2 + (-6)^2) BD = sqrt(36 + 36) BD = sqrt(72) = 6sqrt(2) units Step 2: Calculate the area of the rhombus. The area of a rhombus is given by the formula (1)/(2) d_1 d_2, where d_1 and d_2 are the lengths of the diagonals. Area = (1)/(2) × AC × BD Area = (1)/(2) × (4sqrt(2)) × (6sqrt(2)) Area = (1)/(2) × 24 × (sqrt(2))^2 Area = (1)/(2) × 24 × 2 Area = 24 square units b) Also find the distance between its sides AB and CD. Step 1: Find the equation of line AB. Points A(3, 0) and B(4, 5). Slope m_AB = (5-0)/(4-3) = (5)/(1) = 5. Using the point-slope form y - y_1 = m(x - x_1): y - 0 = 5(x - 3) y = 5x - 15 5x - y - 15 = 0 Step 2: Find the equation of line CD. Points C(-1, 4) and D(-2, -1). Slope m_CD = (-1-4)/(-2-(-1)) = (-5)/(-1) = 5. Using the point-slope form y - y_1 = m(x - x_1): y - 4 = 5(x - (-1)) y - 4 = 5(x + 1) y - 4 = 5x + 5 5x - y + 9 = 0 Step 3: Calculate the distance between the parallel lines AB and CD. The distance between two parallel lines Ax + By + C_1 = 0 and Ax + By + C_2 = 0 is given by d = (|C_1 - C_2|)/(sqrt(A^2 + B^2)). For line AB: 5x - y - 15 = 0 (A=5, B=-1, C_1=-15). For line CD: 5x - y + 9 = 0 (A=5, B=-1, C_2=9). d = (|-15 - 9|)/(sqrt(5^2 + (-1)^2)) d = |