Question

what is the area of the rectangle?
14 units squared
10 units squared
5 units squared
20 units squared

Explanation:

Step1: Calculate the length of side \(AB\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For points \(A(-4,2)\) and \(B(-1,3)\):

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Step2: Calculate the length of side \(BC\)

For points \(B(-1,3)\) and \(C(1,-3)\):

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Step3: Use the area formula for a rectangle \(A = l\times w\)

Here \(l = 2\sqrt{10}\), \(w=\sqrt{10}\), so \(A=\sqrt{10}\times2\sqrt{10}=2\times10 = 20\)

Another method (using the grid - counting and decomposition):

Step1: Enclose the rectangle in a larger rectangle

The larger rectangle has length \(5\) (from \(x=-4\) to \(x = 1\)) and width \(6\) (from \(y=-4\) to \(y = 2\)). Area of the larger rectangle \(A_{large}=5\times6=30\)

Step2: Calculate the area of the four right - angled triangles outside the given rectangle

Two triangles have base \(3\) and height \(1\), and two triangles have base \(2\) and height \(3\)
Area of the four triangles: \(2\times(\frac{1}{2}\times3\times1)+2\times(\frac{1}{2}\times2\times3)=3 + 6=9\)

Step3: Calculate the area of the given rectangle

\(A=30-(3 + 6)=20\)

Answer:

20 units squared