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 — here are the solutions for the triangle ABC with vertices A(2;-1), B(-3;4), and C(4;5). i) Sketch the using the co-ordinates To sketch the triangle, plot the points A(2,-1), B(-3,4), 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,-1) and B(-3,4). AB = sqrt((-3 - 2)^2 + (4 - (-1))^2) AB = sqrt((-5)^2 + (4 + 1)^2) AB = sqrt(25 + (5)^2) AB = sqrt(25 + 25) AB = sqrt(50) AB = sqrt(25 × 2) AB = 5sqrt(2) Step 2: Calculate the length of side BC. Given B(-3,4) and C(4,5). BC = sqrt((4 - (-3))^2 + (5 - 4)^2) BC = sqrt((4 + 3)^2 + (1)^2) BC = sqrt((7)^2 + 1) BC = sqrt(49 + 1) BC = sqrt(50) BC = sqrt(25 × 2) BC = 5sqrt(2) Step 3: Calculate the length of side AC. Given A(2,-1) and C(4,5). AC = sqrt((4 - 2)^2 + (5 - (-1))^2) AC = sqrt((2)^2 + (5 + 1)^2) AC = sqrt(4 + (6)^2) AC = sqrt(4 + 36) AC = sqrt(40) AC = sqrt(4 × 10) AC = 2sqrt(10) The lengths of the sides are: AB = 5sqrt(2), BC = 5sqrt(2), AC = 2sqrt(10) iii) Calculate the perimeter of the The perimeter is the sum of the lengths of all sides. P = AB + BC + AC P = 5sqrt(2) + 5sqrt(2) + 2sqrt(10) P = 10sqrt(2) + 2sqrt(10) To get a numerical value, we approximate sqrt(2) ≈ 1.414 and sqrt(10) ≈ 3.162. P ≈ 10(1.414) + 2(3.162) P ≈ 14.14 + 6.324 P ≈ 20.464 The perimeter of the triangle is: P = 10sqrt(2) + 2sqrt(10) ≈ 20.464 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 = 5sqrt(2) BC = 5sqrt(2) AC = 2sqrt(10) Since AB = BC, two sides of the triangle are equal. Therefore, the triangle is an isosceles triangle. The triangle is isosceles. What's next?