Question

square abcd is graphed in the coordinate plane below. what is the area of square abcd?

Explanation:

Step1: Find the length of the diagonals

The diagonals of the square \(AC\) and \(BD\).
For diagonal \(AC\): The coordinates of \(A(-3,1)\) and \(C(4,1)\). Using the distance formula for points \((x_1,y_1)\) and \((x_2,y_2)\) which is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(y_1 = y_2=1\), so \(AC=\vert- 3-4\vert=7\).
For diagonal \(BD\): The coordinates of \(B(0,5)\) and \(D(0,-3)\). Using the distance formula, here \(x_1 = x_2 = 0\), so \(BD=\vert5-(-3)\vert=8\).

Step2: Use the formula for the area of a square in terms of diagonals

The area of a square \(A=\frac{1}{2}d_1d_2\), where \(d_1\) and \(d_2\) are the lengths of the diagonals.
Substitute \(d_1 = 7\) and \(d_2 = 8\) into the formula: \(A=\frac{1}{2}\times7\times8\).
Calculate \(\frac{1}{2}\times7\times8=28\).

Answer:

\(28\)