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?
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

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

Step2: Calculate the area of the right - triangle

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

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

• In a rectangle, opposite sides are equal and parallel.
• If \(A(1,2)\), \(B(4,2)\), \(C(4,-1)\), for \(ABCD\) to be a rectangle.
• The \(x\) - coordinate of \(D\) is the same as the \(x\) - coordinate of \(A\) (since \(AD\parallel BC\)) and the \(y\) - coordinate of \(D\) is the same as the \(y\) - coordinate of \(C\) (since \(CD\parallel AB\)).
• So, \(D(1,-1)\).
• Check the lengths:
• \(AD=\sqrt{(1 - 1)^2+(-1 - 2)^2}=3\), \(CD=\sqrt{(1 - 4)^2+(-1+1)^2}=3\).
• \(AB = 3\), \(BC = 3\), \(AD = 3\), \(CD = 3\) and \(AB\parallel CD\), \(BC\parallel AD\)

Answer:

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