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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Here are the solutions for the triangle ABC with vertices A(-2;-3), B(-4;1), and C(4;5). i) Sketch the using the co-ordinates To sketch the triangle, plot the points A(-2,-3), B(-4,1), and C(4,5) on a Cartesian coordinate plane. Then, connect the points with straight lines to form the triangle ABC. ii) Calculate the distance of each side We use the distance formula: d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). Step 1: Calculate the length of side AB. Given A(-2,-3) and B(-4,1). AB = sqrt((-4 - (-2))^2 + (1 - (-3))^2) AB = sqrt((-4 + 2)^2 + (1 + 3)^2) AB = sqrt((-2)^2 + (4)^2) AB = sqrt(4 + 16) AB = sqrt(20) AB = sqrt(4 × 5) AB = 2sqrt(5) Step 2: Calculate the length of side BC. Given B(-4,1) and C(4,5). BC = sqrt((4 - (-4))^2 + (5 - 1)^2) BC = sqrt((4 + 4)^2 + (4)^2) BC = sqrt((8)^2 + (4)^2) BC = sqrt(64 + 16) BC = sqrt(80) BC = sqrt(16 × 5) BC = 4sqrt(5) Step 3: Calculate the length of side AC. Given A(-2,-3) and C(4,5). AC = sqrt((4 - (-2))^2 + (5 - (-3))^2) AC = sqrt((4 + 2)^2 + (5 + 3)^2) AC = sqrt((6)^2 + (8)^2) AC = sqrt(36 + 64) AC = sqrt(100) AC = 10 The lengths of the sides are: AB = 2sqrt(5), BC = 4sqrt(5), AC = 10 iii) Calculate the perimeter of the The perimeter is the sum of the lengths of all sides. P = AB + BC + AC P = 2sqrt(5) + 4sqrt(5) + 10 P = 6sqrt(5) + 10 To get a numerical value, we approximate sqrt(5) ≈ 2.236. P ≈ 6(2.236) + 10 P ≈ 13.416 + 10 P ≈ 23.416 The perimeter of the triangle is: P = 6sqrt(5) + 10 ≈ 23.416 iv) Determine whether the is equilateral, isosceles or scalene • An equilateral triangle has all three sides equal. • An isosceles triangle has at least two sides equal. • A scalene triangle has all three sides of different lengths. Comparing the side lengths: AB = 2sqrt(5) BC = 4sqrt(5) AC = 10 Since 2sqrt(5) ≠ 4sqrt(5) ≠ 10, all three sides have different lengths. Therefore, the triangle is a scalene triangle. The triangle is scalene. 3 done, 2 left today. You're making progress.