Nathan draws a triangle on the coordinate grid. He marks his points at (1,5), (4,1), and (−4,−5). What is the perimeter of the triangle, rounded to the nearest tenth of a unit? Show all steps that Nathan must take to find the solution. in 1 paragraph

Answer 1

To find the perimeter of the triangle formed by the points (1,5), (4,1), and (−4,−5), Nathan first calculates the lengths of the three sides using the distance formula, which is given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For side AB between points A(1,5) and B(4,1), he calculates \(AB = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\). For side BC between points B(4,1) and C(-4,-5), he calculates \(BC = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10\). Finally, for side AC between points A(1,5) and C(-4,-5), he calculates \(AC = \sqrt{(-4 - 1)^2 + (-5 - 5)^2} = \sqrt{(-5)^2 + (-10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \approx 11.2\). Adding these lengths together, Nathan finds the perimeter: \(P = AB + BC + AC = 5 + 10 + 11.2 \approx 26.2\). Thus, the perimeter of the triangle, rounded to the nearest tenth of a unit, is approximately 26.2.