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 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

• Calculate the lengths of the sides.
• The distance between \(A(1,2)\) and \(B(4,2)\) using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 1,y_1=2,x_2 = 4,y_2 = 2\), so \(AB=\sqrt{(4 - 1)^2+(2 - 2)^2}=\sqrt{9+0}=3\).
• The distance between \(B(4,2)\) and \(C(4,-1)\). Here, \(x_1 = 4,y_1=2,x_2 = 4,y_2=-1\), so \(BC=\sqrt{(4 - 4)^2+(-1 - 2)^2}=\sqrt{0 + 9}=3\).
• The slope of \(AB\): \(m_{AB}=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2-2}{4 - 1}=0\) (horizontal line).
• The slope of \(BC\): \(m_{BC}=\frac{-1 - 2}{4 - 4}\), undefined (vertical line).
• Since \(AB\perp BC\) and there are only three points, the figure is a right - triangle.

Step2: Calculate the area of the right - triangle

• For a right - triangle with legs \(a\) and \(b\), the area formula is \(A=\frac{1}{2}ab\).
• Here, \(a = AB = 3\) and \(b=BC = 3\).
• 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 \(AB\) is parallel to \(CD\) and \(BC\) is parallel to \(AD\).
• Since \(AB\) is horizontal (\(y = 2\) for \(AB\)) and \(BC\) is vertical (\(x = 4\) for \(BC\)).
• Let the coordinates of \(D\) be \((x,y)\).
• Since \(AB\parallel CD\), \(y=-1\) (same \(y\) - value as \(C\) for the horizontal line parallel to \(AB\)).
• Since \(BC\parallel AD\), \(x = 1\) (same \(x\) - value as \(A\) for the vertical line parallel to \(BC\)). So \(D=(1,-1)\).
• Check the lengths: \(AD=\sqrt{(1 - 1)^2+(-1 - 2)^2}=3\), \(CD=\sqrt{(1 - 4)^2+(-1+1)^2}=3\), \(AB = 3\), \(BC = 3\). Also, slopes: \(m_{AD}\) is undefined (\(x = 1\) for \(AD\)), \(m_{CD}=0\) (\(y=-1\) for \(CD\)).

Answer:

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