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 slopes of \(AB\), \(BC\).
• The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
• For \(A(1,2)\) and \(B(4,2)\), \(m_{AB}=\frac{2 - 2}{4 - 1}=0\) (horizontal line).
• For \(B(4,2)\) and \(C(4,-1)\), \(m_{BC}=\frac{-1 - 2}{4 - 4}\) (undefined, vertical line).
• Since \(AB\) is horizontal and \(BC\) is vertical, \(\angle ABC = 90^{\circ}\), and the figure is a right - triangle.

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

• Use the formula for the area of a right - triangle \(A=\frac{1}{2}\times base\times height\).
• The length of \(AB\): \(d_{AB}=\sqrt{(4 - 1)^2+(2 - 2)^2}=3\) (using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), or since \(y\) - values are the same, \(|4 - 1| = 3\)).
• The length of \(BC\): \(d_{BC}=\sqrt{(4 - 4)^2+(-1 - 2)^2}=3\) (using the distance formula, or since \(x\) - values are the same, \(| - 1-2| = 3\)).
• \(A=\frac{1}{2}\times AB\times BC=\frac{1}{2}\times3\times3=\frac{9}{2}\)

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

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

Answer:

1. The shape is a right - triangle.
2. The area is \(\frac{9}{2}\) square units.
3. The coordinates of \(D\) are \((1,-1)\)