Question

suppose a rectangle is drawn on a coordinate plane. what is the perimeter of the rectangle if its vertices are at coo (0, 0), (4, 4), (8, 3), and (1, -1)? round your answer to the nearest tenth. (1 point) o 14.1 units o 5.6 units o 22.8 units o 8 units

Explanation:

Step1: Identify two adjacent vertices

Let's take two adjacent vertices, say \((0, 0)\) and \((4, 4)\), and another pair \((0, 0)\) and \((1, -1)\) (wait, actually, we need to find the length and width. Let's use the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). First, find the length of two adjacent sides. Let's take \((0, 0)\) and \((4, 4)\): \(d_1=\sqrt{(4 - 0)^2+(4 - 0)^2}=\sqrt{16 + 16}=\sqrt{32}\approx5.656\). Then take \((0, 0)\) and \((1, -1)\)? Wait, no, maybe the vertices are \((0,0)\), \((4,4)\), \((5,3)\), \((1, -1)\)? Wait, maybe I misread. Wait, the vertices are \((0,0)\), \((4,4)\), \((5,3)\), \((1, -1)\)? Wait, no, let's check the distance between \((0,0)\) and \((4,4)\): \(d=\sqrt{(4 - 0)^2+(4 - 0)^2}=\sqrt{32}\approx5.656\). Then between \((4,4)\) and \((5,3)\): \(d=\sqrt{(5 - 4)^2+(3 - 4)^2}=\sqrt{1 + 1}=\sqrt{2}\approx1.414\). No, that's not a rectangle. Wait, maybe the vertices are \((0,0)\), \((4,4)\), \((5,3)\), \((1, -1)\)? Wait, maybe I made a mistake. Wait, let's re - examine. Wait, the problem says "rectangle", so opposite sides are equal and diagonals are equal. Let's find the distance between \((0,0)\) and \((5,3)\): \(d=\sqrt{(5 - 0)^2+(3 - 0)^2}=\sqrt{25 + 9}=\sqrt{34}\approx5.83\). Between \((4,4)\) and \((1, -1)\): \(d=\sqrt{(1 - 4)^2+(-1 - 4)^2}=\sqrt{9 + 25}=\sqrt{34}\approx5.83\). Then between \((0,0)\) and \((4,4)\): \(\sqrt{32}\approx5.656\), between \((5,3)\) and \((1, -1)\): \(\sqrt{(1 - 5)^2+(-1 - 3)^2}=\sqrt{16 + 16}=\sqrt{32}\approx5.656\). So the length \(l=\sqrt{34}\approx5.83\), width \(w = \sqrt{32}\approx5.656\)? Wait, no, that can't be. Wait, maybe the vertices are \((0,0)\), \((4,4)\), \((5,3)\), \((1, -1)\). Wait, the perimeter of a rectangle is \(P = 2(l + w)\). Let's calculate \(l\) and \(w\) correctly. Let's take two adjacent vertices: \((0,0)\) and \((4,4)\): \(d_1=\sqrt{(4 - 0)^2+(4 - 0)^2}=\sqrt{32}\approx5.656\). Then \((4,4)\) and \((5,3)\): \(d_2=\sqrt{(5 - 4)^2+(3 - 4)^2}=\sqrt{2}\approx1.414\). No, that's not a rectangle. Wait, maybe I misread the vertices. Wait, the problem says "vertices are at \((0,0)\), \((4,4)\), \((5,3)\), and \((1, -1)\)? Wait, maybe the correct vertices are \((0,0)\), \((4,0)\), \((4,4)\), \((0,4)\)? No, the given vertices are \((0,0)\), \((4,4)\), \((5,3)\), \((1, -1)\). Wait, let's check the slopes. The slope between \((0,0)\) and \((4,4)\) is \(\frac{4 - 0}{4 - 0}=1\). The slope between \((5,3)\) and \((1, -1)\) is \(\frac{-1 - 3}{1 - 5}=\frac{-4}{-4}=1\). The slope between \((0,0)\) and \((1, -1)\) is \(\frac{-1 - 0}{1 - 0}=-1\). The slope between \((4,4)\) and \((5,3)\) is \(\frac{3 - 4}{5 - 4}=-1\). So it is a rectangle, with length between \((0,0)\) and \((1, -1)\): \(d=\sqrt{(1 - 0)^2+(-1 - 0)^2}=\sqrt{2}\approx1.414\)? No, that's not. Wait, no, the length between \((0,0)\) and \((4,4)\) is \(\sqrt{(4 - 0)^2+(4 - 0)^2}=\sqrt{32}\approx5.656\), and the length between \((0,0)\) and \((1, -1)\) is \(\sqrt{2}\approx1.414\). Wait, no, the distance between \((0,0)\) and \((5,3)\) is \(\sqrt{25 + 9}=\sqrt{34}\approx5.83\), and between \((4,4)\) and \((1, -1)\) is also \(\sqrt{34}\approx5.83\). The distance between \((0,0)\) and \((4,4)\) is \(\sqrt{32}\approx5.656\), and between \((5,3)\) and \((1, -1)\) is also \(\sqrt{32}\approx5.656\). So the length \(l=\sqrt{34}\approx5.83\), width \(w=\sqrt{32}\approx5.656\)? Wait, no, that's not right. Wait, perimeter \(P = 2(l + w)\). Let's calculate \(l\) as the distance between \((0,0)\) and \((5,3)\): \(\sqrt{(5 - 0)^2+(3 - 0)^2}=\sqrt{25 + 9}=\sqrt{34}\approx5.83\). \(w\) as the distance between \((0,0)\) and \((4,4)\): \(\sqrt{(4 - 0)^2+(4 - 0)^2}=\sqrt{32}\approx5.656\). Then \(P = 2(5.83+5.656)=2(11.486)=22.972\approx23\), but the options are 14.1, 8.6, 22.8, 8. Wait, maybe I misread the vertices. Wait, maybe the vertices are \((0,0)\), \((4,0)\), \((4,4)\), \((0,4)\)? No, the given vertices are \((0,0)\), \((4,4)\), \((5,3)\), \((1, -1)\). Wait, maybe the correct vertices are \((0,0)\), \((4,0)\), \((4,3)\), \((0,3)\)? No. Wait, let's recalculate the distance between \((0,0)\) and \((4,4)\): \(\sqrt{(4 - 0)^2+(4 - 0)^2}=\sqrt{32}\approx5.656\). Distance between \((4,4)\) and \((5,3)\): \(\sqrt{(5 - 4)^2+(3 - 4)^2}=\sqrt{2}\approx1.414\). No, that's not a rectangle. Wait, maybe the vertices are \((0,0)\), \((4,4)\), \((5,3)\), \((1, -1)\) is a typo, and it should be \((0,0)\), \((4,0)\), \((4,3)\), \((0,3)\). Then length is 4, width is 3, perimeter is \(2(4 + 3)=14\), close to 14.1. Alternatively, maybe the vertices are \((0,0)\), \((4,4)\), \((5,3)\), \((1, -1)\). Let's calculate the distance between \((0,0)\) and \((1, -1)\): \(\sqrt{(1 - 0)^2+(-1 - 0)^2}=\sqrt{2}\approx1.414\). Distance between \((1, -1)\) and \((5,3)\): \(\sqrt{(5 - 1)^2+(3 + 1)^2}=\sqrt{16 + 16}=\sqrt{32}\approx5.656\). Distance between \((5,3)\) and \((4,4)\): \(\sqrt{(4 - 5)^2+(4 - 3)^2}=\sqrt{2}\approx1.414\). Distance between \((4,4)\) and \((0,0)\): \(\sqrt{32}\approx5.656\). So the length is \(\sqrt{32}\approx5.656\), width is \(\sqrt{2}\approx1.414\). Then perimeter is \(2(5.656 + 1.414)=2(7.07)=14.14\approx14.1\). Ah, that's one of the options. So the perimeter is approximately 14.1 units.

Step1: Calculate length of one side

Take vertices \((0,0)\) and \((4,4)\). Using distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(d_1=\sqrt{(4 - 0)^2+(4 - 0)^2}=\sqrt{16 + 16}=\sqrt{32}\approx5.656\).

Step2: Calculate length of adjacent side

Take vertices \((0,0)\) and \((1, -1)\). Using distance formula, \(d_2=\sqrt{(1 - 0)^2+(-1 - 0)^2}=\sqrt{1 + 1}=\sqrt{2}\approx1.414\).

Step3: Calculate perimeter

Perimeter of rectangle \(P = 2(d_1 + d_2)\). Substitute \(d_1\approx5.656\) and \(d_2\approx1.414\): \(P = 2(5.656+1.414)=2(7.07)=14.14\approx14.1\).

Answer:

14.1 units