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 \(C\)

• Calculate the lengths of \(AB\) and \(BC\) using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• For \(AB\): \(x_1 = 1,y_1 = 2,x_2 = 4,y_2 = 2\). Then \(AB=\sqrt{(4 - 1)^2+(2 - 2)^2}=\sqrt{9+0}=3\).
• For \(BC\): \(x_1 = 4,y_1 = 2,x_2 = 4,y_2=-1\). Then \(BC=\sqrt{(4 - 4)^2+(-1 - 2)^2}=\sqrt{0 + 9}=3\).
• Since \(AB\) is horizontal (\(y\)-coordinates are the same) and \(BC\) is vertical (\(x\)-coordinates are the same), the figure formed by \(A\), \(B\), and \(C\) is a right - triangle.

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

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

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 \(A=(1,2)\), \(B=(4,2)\), \(C=(4,-1)\), then for \(ABCD\) to be a rectangle, the \(x\)-coordinate of \(D\) should be the same as the \(x\)-coordinate of \(A\) (because \(AD\) should be parallel to \(BC\)) and the \(y\)-coordinate of \(D\) should be the same as the \(y\)-coordinate of \(C\) (because \(CD\) should be parallel to \(AB\)). So \(D=(1,-1)\).

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 \(D\) such that \(ABCD\) is a rectangle are \((1,-1)\).