Question

finding area draw the figure in a coordinate plane and find its area. 33. square defined by w(5,0), x(0,5), y(-5,0), and z(0,-5)

Explanation:

Step1: Identify the side length

First, we can use the distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) which is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Let's find the distance between \(W(5,0)\) and \(X(0,5)\).
\(d=\sqrt{(0 - 5)^2+(5 - 0)^2}=\sqrt{(- 5)^2+5^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}\)

Step2: Calculate the area of the square

The area of a square is given by \(A = s^2\), where \(s\) is the side length. We found \(s = 5\sqrt{2}\), so the area \(A=(5\sqrt{2})^2\).
\(A = 25\times2=50\)

Alternatively, we can use the formula for the area of a square using the coordinates. The diagonals of the square can be found. The distance between \(W(5,0)\) and \(Y(- 5,0)\) is \(d_1=\sqrt{(-5 - 5)^2+(0 - 0)^2}=\sqrt{(-10)^2}=10\), and the distance between \(X(0,5)\) and \(Z(0,-5)\) is \(d_2=\sqrt{(0 - 0)^2+(-5 - 5)^2}=\sqrt{(-10)^2}=10\). For a square, the area is also \(\frac{1}{2}\times d_1\times d_2\) (since the diagonals of a square are equal and perpendicular, and area of a rhombus is \(\frac{1}{2}\times d_1\times d_2\), and square is a rhombus). So \(A=\frac{1}{2}\times10\times10 = 50\)

Answer:

The area of the square is \(\boxed{50}\)