Question

use the coordinates to compute the exact area of the rectangle(1 point)$10\sqrt{10}units^{2}$$5\sqrt{10}units^{2}$$30units^{2}$$60units^{2}$

Explanation:

Step1: Calculate the length of one side

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Let's find the distance between two adjacent points, say \(A(-3,5)\) and the next vertex (assume the next vertex has coordinates \((- 2,3)\) - from the graph structure, but more accurately, if we consider the side between \(A(-3,5)\) and the mid - connecting point (using the rectangle property). Wait, better approach: use the formula for the area of a parallelogram (since rectangle is a parallelogram) \(A=\vert\vec{u}\times\vec{v}\vert\). Let \(\vec{u}=(3 - (-3),7 - 5)=(6,2)\) and \(\vec{v}=(6-0,-2 - (-4))=(6,2)\). Wait no, wrong. Wait, another approach: the length of one side \(l_1\): between \(A(-3,5)\) and \(B(3,7)\). \(d_1=\sqrt{(3+3)^2+(7 - 5)^2}=\sqrt{36 + 4}=\sqrt{40}=2\sqrt{10}\). The length of the adjacent side \(l_2\): between \(B(3,7)\) and \(C(6,-2)\). \(d_2=\sqrt{(6 - 3)^2+(-2 - 7)^2}=\sqrt{9+81}=\sqrt{90}=3\sqrt{10}\). But wait, no, rectangle area \(A = l\times w\). Wait, another way: use the shoelace formula. For points \(A(-3,5)\), \(B(3,7)\), \(C(6,-2)\), \(D(0,-4)\) (in order). The shoelace formula \(A=\frac{1}{2}\vert\sum_{i = 1}^{n}(x_iy_{i + 1}-x_{i+1}y_i)\vert\), where \(x_5=x_1\), \(y_5=y_1\). \(A=\frac{1}{2}\vert(-3\times7+3\times(-2)+6\times(-4)+0\times5)- (5\times3 + 7\times6+(-2)\times0+(-4)\times(-3))\vert\). \(=\frac{1}{2}\vert(-21-6 - 24+0)-(15 + 42+0 + 12)\vert=\frac{1}{2}\vert(-51)-(69)\vert=\frac{1}{2}\vert-120\vert = 60\).

Answer:

\(60\ units^{2}\)