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 of this figure?
2. calculate the area of the figure formed by points a, b, and c.
3. 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:

Step1: Determine the shape of the figure formed by points \(A\), \(B\), and \(C\)

Plot the points \(A(1,2)\), \(B(4,2)\), and \(C(4, - 1)\) on a coordinate grid.
The slope of \(AB\) is \(m_{AB}=\frac{2 - 2}{4 - 1}=0\) (horizontal line).
The slope of \(BC\) is \(m_{BC}=\frac{-1 - 2}{4 - 4}\), which is undefined (vertical line).
Since \(AB\) and \(BC\) are perpendicular, the figure formed by \(A\), \(B\), and \(C\) is a right - triangle.

Step2: Calculate the area of the right - triangle \(ABC\)

The length of \(AB\) (base) is \(|4 - 1|=3\) (using the distance formula for horizontal points \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and since \(y_1 = y_2\), \(d=x_2 - x_1\)).
The length of \(BC\) (height) is \(| - 1-2| = 3\) (using the distance formula for vertical points \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and since \(x_1 = x_2\), \(d=y_2 - y_1\)).
The area of a triangle is \(A=\frac{1}{2}\times base\times height\).
Substitute base \(b = 3\) and height \(h = 3\) into the formula: \(A=\frac{1}{2}\times3\times3=\frac{9}{2}=4.5\).

Step3: Find the coordinates of point \(D\) to form rectangle \(ABCD\)

In a rectangle, opposite sides are equal and parallel.
Since \(AB\) is horizontal with length \(3\) and \(BC\) is vertical with length \(3\).
If \(A(1,2)\), \(B(4,2)\), \(C(4, - 1)\), for \(ABCD\) to be a rectangle, the \(x\) - coordinate of \(D\) is \(1\) (same \(x\) - coordinate as \(A\)) and the \(y\) - coordinate of \(D\) is \(-1\) (same \(y\) - coordinate as \(C\)). So \(D(1,-1)\).
Check the lengths:
\(AB=\sqrt{(4 - 1)^2+(2 - 2)^2}=3\), \(CD=\sqrt{(1 - 4)^2+(-1+1)^2}=3\)
\(BC=\sqrt{(4 - 4)^2+(-1 - 2)^2}=3\), \(AD=\sqrt{(1 - 1)^2+(-1 - 2)^2}=3\)
\(AB\parallel CD\) (both have slope \(0\)) and \(BC\parallel AD\) (both have undefined slope).

Answer:

1. The shape of the figure formed by \(A\), \(B\), and \(C\) is a right - triangle.
2. The area of the figure (right - triangle \(ABC\)) is \(4.5\).
3. The coordinates of point \(D\) is \((1,-1)\).