Question

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?

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:

Step1: Determine the shape of triangle \(ABC\)

• Calculate the lengths of the sides:
• The distance between \(A(1,2)\) and \(B(4,2)\) is \(AB=\sqrt{(4 - 1)^2+(2 - 2)^2}=\sqrt{3^2}=3\) (using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)).
• The distance between \(B(4,2)\) and \(C(4,-1)\) is \(BC=\sqrt{(4 - 4)^2+(-1 - 2)^2}=\sqrt{(-3)^2}=3\).
• The distance between \(A(1,2)\) and \(C(4,-1)\) is \(AC=\sqrt{(4 - 1)^2+(-1 - 2)^2}=\sqrt{3^2+(-3)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\).
• Since \(AB^{2}+BC^{2}=3^{2}+3^{2}=9 + 9 = 18=(3\sqrt{2})^{2}=AC^{2}\), by the Pythagorean theorem, \(\triangle ABC\) is a right - triangle.

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

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

Step3: Find the coordinates of point \(D\)

• In a rectangle, opposite sides are equal and parallel.
• If \(ABCD\) is a rectangle, since \(AB\) is horizontal (\(y\) - coordinate is constant for \(A\) and \(B\)) and \(BC\) is vertical (\(x\) - coordinate is constant for \(B\) and \(C\)).
• The \(x\) - coordinate of \(D\) should be the same as the \(x\) - coordinate of \(A\) (\(x = 1\)), and the \(y\) - coordinate of \(D\) should be the same as the \(y\) - coordinate of \(C\) (\(y=-1\)). So \(D=(1,-1)\).
• Check the side lengths: \(AB = 3\), \(BC = 3\), \(CD=\sqrt{(1 - 4)^2+(-1+1)^2}=3\), \(DA=\sqrt{(1 - 1)^2+(2 + 1)^2}=3\). Also, \(AB\parallel CD\) (both have slope \(m = 0\)) and \(BC\parallel DA\) (both have an undefined slope for vertical lines).

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 \((1,-1)\).