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 \(BC\).
• For \(AB\): Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(A(1,2)\) and \(B(4,2)\), \(d_{AB}=\sqrt{(4 - 1)^2+(2 - 2)^2}=\sqrt{3^2+0^2}=3\).
• For \(BC\): For \(B(4,2)\) and \(C(4,-1)\), \(d_{BC}=\sqrt{(4 - 4)^2+(-1 - 2)^2}=\sqrt{0^2+(-3)^2}=3\).
• Check the slopes of \(AB\) and \(BC\).
• Slope of \(AB\): \(m_{AB}=\frac{2 - 2}{4 - 1}=0\) (horizontal line).
• Slope of \(BC\): \(m_{BC}=\frac{-1 - 2}{4 - 4}\), undefined (vertical line).
• Since \(AB\perp BC\) (product of slopes \(m_{AB}\times m_{BC}=0\times\) undefined (perpendicular as one is horizontal and one is vertical)), the figure \(ABC\) 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 \(ABCD\) is a rectangle, from the properties of rectangles:
• The \(x\) - coordinate of \(D\) is the same as the \(x\) - coordinate of \(A\) (since \(AD\parallel BC\) and \(AD = BC\)), and the \(y\) - coordinate of \(D\) is the same as the \(y\) - coordinate of \(C\) (since \(CD\parallel AB\) and \(CD = AB\)).
• So, \(D(1,-1)\)

Answer:

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