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:

Sub - question 1

Step 1: Analyze the coordinates

For point \(A(1,2)\), \(B(4,2)\), and \(C(4, - 1)\). The \(y\) - coordinate of \(A\) and \(B\) is the same (\(y = 2\)), so the line segment \(AB\) is horizontal. The \(x\) - coordinate of \(B\) and \(C\) is the same (\(x = 4\)), so the line segment \(BC\) is vertical.

Step 2: Determine the angle between \(AB\) and \(BC\)

Since \(AB\) is horizontal (slope \(m_{AB}=\frac{2 - 2}{4 - 1}=0\)) and \(BC\) is vertical (slope \(m_{BC}=\frac{-1 - 2}{4 - 4}\) is undefined), the angle between \(AB\) and \(BC\) is \(90^{\circ}\). Also, we have three points, so the figure is a right - angled triangle.

Step 1: Calculate the length of \(AB\)

The length of a horizontal line segment with endpoints \((x_1,y)\) and \((x_2,y)\) is given by \(|x_2 - x_1|\). For \(A(1,2)\) and \(B(4,2)\), \(AB=|4 - 1| = 3\).

Step 2: Calculate the length of \(BC\)

The length of a vertical line segment with endpoints \((x,y_1)\) and \((x,y_2)\) is given by \(|y_2 - y_1|\). For \(B(4,2)\) and \(C(4,-1)\), \(BC=| - 1-2|=| - 3| = 3\)? Wait, no. Wait, \(y_1 = 2\) and \(y_2=-1\), so \(BC = |2-(-1)|=3\)? Wait, no, the formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), but since it's vertical, \(x_2=x_1 = 4\), so \(BC=\sqrt{(4 - 4)^2+(-1 - 2)^2}=\sqrt{0 + 9}=3\). Wait, no, \(A(1,2)\), \(B(4,2)\), \(C(4,-1)\). The base of the right - triangle can be \(AB\) with length \(4 - 1=3\) and the height can be \(2-(-1)=3\)? Wait, no, the vertical distance from \(B\) to \(C\) is \(2-(-1) = 3\). The area of a right - triangle is \(A=\frac{1}{2}\times\text{base}\times\text{height}\). Here, base \(AB = 3\) (horizontal distance between \(A\) and \(B\)) and height \(BC = 3\) (vertical distance between \(B\) and \(C\))? Wait, no, \(AB\) is from \(x = 1\) to \(x = 4\) (length 3) and \(BC\) is from \(y = 2\) to \(y=-1\) (length 3). So the area of the right - triangle \(A=\frac{1}{2}\times AB\times BC=\frac{1}{2}\times3\times3 = 4.5\)? Wait, no, wait: \(A(1,2)\), \(B(4,2)\), \(C(4,-1)\). The horizontal side \(AB\) has length \(4 - 1=3\), the vertical side \(BC\) has length \(2-(-1)=3\). So the area of the right - triangle is \(\frac{1}{2}\times3\times3 = 4.5\) square units.

Step 1: Identify the base and height

The base of the right - triangle is the length of \(AB\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(A(1,2)\) and \(B(4,2)\), \(AB=\sqrt{(4 - 1)^2+(2 - 2)^2}=\sqrt{9 + 0}=3\).
The height of the right - triangle is the length of \(BC\). For \(B(4,2)\) and \(C(4,-1)\), \(BC=\sqrt{(4 - 4)^2+(-1 - 2)^2}=\sqrt{0 + 9}=3\).

Step 2: Calculate the area

The area of a right - triangle is given by \(A=\frac{1}{2}\times\text{base}\times\text{height}\). Substituting the values of base \(AB = 3\) and height \(BC = 3\) into the formula, we get \(A=\frac{1}{2}\times3\times3=\frac{9}{2}=4.5\) square units.

Step 1: Recall the properties of a rectangle

In a rectangle \(ABCD\), the opposite sides are equal and parallel, and the adjacent sides are perpendicular. Also, the vector \(\overrightarrow{AB}=\overrightarrow{DC}\) and \(\overrightarrow{AD}=\overrightarrow{BC}\).
We know that \(A(1,2)\), \(B(4,2)\), \(C(4,-1)\). The vector \(\overrightarrow{AB}=(4 - 1,2 - 2)=(3,0)\) and the vector \(\overrightarrow{BC}=(4 - 4,-1 - 2)=(0,-3)\).
To find point \(D\), we can use the fact that in a rectangle \(ABCD\), \(D = A+\overrightarrow{BC}\) (or \(D = C-\overrightarrow{AB}\)).
If we consider the coordinates: since \(AB\) is horizontal from \(A(1,2)\) to \(B(4,2)\) (change in \(x\) is \(3\), change in \(y\) is \(0\)) and \(BC\) is vertical from \(B(4,2)\) to \(C(4,-1)\) (change in \(x\) is \(0\), change in \(y\) is \(- 3\)).
For a rectangle \(ABCD\), the coordinates of \(D\) can be found as follows: The \(x\) - coordinate of \(D\) should be the same as the \(x\) - coordinate of \(A\) (because \(AD\) should be parallel to \(BC\), and \(BC\) is vertical, so \(AD\) is vertical) and the \(y\) - coordinate of \(D\) should be the same as the \(y\) - coordinate of \(C\) (because \(CD\) should be parallel to \(AB\), and \(AB\) is horizontal, so \(CD\) is horizontal).
So, the \(x\) - coordinate of \(D\) is \(1\) (same as \(A\)'s \(x\) - coordinate) and the \(y\) - coordinate of \(D\) is \(-1\) (same as \(C\)'s \(y\) - coordinate). So \(D=(1,-1)\).

Step 2: Verify the rectangle

• Check the sides:
• \(AB\): length is \(3\) (from \(A(1,2)\) to \(B(4,2)\)). \(CD\): from \(C(4,-1)\) to \(D(1,-1)\), length is \(|4 - 1| = 3\). So \(AB = CD\).
• \(BC\): length is \(3\) (from \(B(4,2)\) to \(C(4,-1)\)). \(AD\): from \(A(1,2)\) to \(D(1,-1)\), length is \(|2-(-1)| = 3\). So \(BC = AD\).
• Check the angles: \(AB\) is horizontal (slope \(0\)) and \(BC\) is vertical (undefined slope), so \(\angle ABC = 90^{\circ}\). Similarly, \(AB\parallel CD\) (both horizontal) and \(BC\parallel AD\) (both vertical), so all angles are \(90^{\circ}\).

Answer:

The shape of the figure is a right - angled triangle.

Sub - question 2