Question

in the coordinate plane, points a, b, and c have coordinates (1,2), (4,2), and (4,−1) respectively.

1. plot points a, b, and c on a coordinate grid, then connect them to form a figure. what is the shape this figure?

1. calculate the area of the figure formed by points a, b, and c.

1. find the coordinates of point d such that quadrilateral abcd is a rectangle. plot point d and verify the rectangle by checking the properties of its sides.

Explanation:

1. Determining the shape

Step1: Analyze the coordinates

For points \(A(1,2)\), \(B(4,2)\), \(C(4, - 1)\). The \(y\) - coordinate of \(A\) and \(B\) is the same (\(y = 2\)), so \(AB\) is a horizontal line. The \(x\) - coordinate of \(B\) and \(C\) is the same (\(x = 4\)), so \(BC\) is a vertical line. A figure with one horizontal and one vertical side meeting at a right - angle is a right - triangle.

2. Calculating the area

Step1: Find the lengths of the legs

The length of \(AB\) is calculated using the distance formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) on a horizontal line (\(y_1=y_2\)): \(d=\vert x_2 - x_1\vert\). For \(A(1,2)\) and \(B(4,2)\), \(AB=\vert4 - 1\vert=3\).
The length of \(BC\) is calculated using the distance formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) on a vertical line (\(x_1 = x_2\)): \(d=\vert y_2 - y_1\vert\). For \(B(4,2)\) and \(C(4,-1)\), \(BC=\vert-1 - 2\vert=3\).

Step2: Apply the area formula for a right - triangle

The area formula for a right - triangle is \(A=\frac{1}{2}\times base\times height\). Here, \(base = AB = 3\) and \(height=BC = 3\). So \(A=\frac{1}{2}\times3\times3=\frac{9}{2}=4.5\)

3. Finding the coordinates of point \(D\)

Step1: Use the properties of a rectangle

In a rectangle, opposite sides are equal and parallel. If \(AB\) is parallel to \(CD\) and \(BC\) is parallel to \(AD\).
The \(x\) - coordinate of \(D\) should be the same as the \(x\) - coordinate of \(A\) (since \(AD\parallel BC\)) and the \(y\) - coordinate of \(D\) should be the same as the \(y\) - coordinate of \(C\) (since \(CD\parallel AB\)). So the coordinates of \(D\) are \((1,-1)\)

Step2: Check the side lengths

\(AB=\vert4 - 1\vert = 3\), \(CD=\vert4 - 1\vert=3\), \(BC=\vert-1 - 2\vert=3\), \(AD=\vert-1 - 2\vert=3\). Also, \(AB\parallel CD\) (both horizontal) and \(BC\parallel AD\) (both vertical)

Answer:

1. The shape is a right - triangle.
2. The area of the figure is \(4.5\)
3. The coordinates of point \(D\) are \((1,-1)\)