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

Step1: Analyze coordinates

For point \(A(1,2)\), \(B(4,2)\), the \(y\) - coordinates are the same (\(y = 2\)), so the line segment \(AB\) is horizontal. The length of \(AB\) is \(|4 - 1|=3\). For point \(B(4,2)\) and \(C(4,-1)\), the \(x\) - coordinates are the same (\(x = 4\)), so the line segment \(BC\) is vertical. The length of \(BC\) is \(|2-(-1)| = 3\). The angle between \(AB\) (horizontal) and \(BC\) (vertical) is \(90^{\circ}\).

Step2: Determine the shape

Since we have three points, and two sides ( \(AB\) and \(BC\)) are perpendicular and of equal length, the figure formed by \(A\), \(B\), \(C\) is a right - angled triangle.

Step1: Recall the formula for the area of a right - angled triangle

The formula for the area of a right - angled triangle is \(A=\frac{1}{2}\times base\times height\).

Step2: Identify base and height

From the coordinates, the length of \(AB\) (base) is \(4 - 1=3\) (since \(A(1,2)\) and \(B(4,2)\)) and the length of \(BC\) (height) is \(2-(-1) = 3\) (since \(B(4,2)\) and \(C(4,-1)\)).

Step3: Calculate the area

Substitute the values of base and height into the formula: \(A=\frac{1}{2}\times3\times3=\frac{9}{2} = 4.5\).

Step1: Recall the properties of a rectangle

In a rectangle \(ABCD\), the opposite sides are equal and parallel. We know that \(AB\parallel CD\) and \(AD\parallel BC\). Since \(AB\) is horizontal (from \(A(1,2)\) to \(B(4,2)\)), \(CD\) should also be horizontal. Since \(BC\) is vertical (from \(B(4,2)\) to \(C(4,-1)\)), \(AD\) should also be vertical.

Step2: Find the coordinates of \(D\)

For \(AD\) to be vertical (same \(x\) - coordinate as \(A\)) and \(CD\) to be horizontal (same \(y\) - coordinate as \(C\)). The \(x\) - coordinate of \(D\) should be the same as the \(x\) - coordinate of \(A\) (i.e., \(x = 1\)) and the \(y\) - coordinate of \(D\) should be the same as the \(y\) - coordinate of \(C\) (i.e., \(y=-1\)). So, the coordinates of \(D\) are \((1,-1)\).

Step3: Verify the rectangle

• Check \(AB\) and \(CD\): The coordinates of \(A(1,2)\), \(B(4,2)\), \(C(4,-1)\), \(D(1,-1)\). The vector \(\overrightarrow{AB}=(4 - 1,2 - 2)=(3,0)\) and \(\overrightarrow{CD}=(1 - 4,-1-(-1))=(-3,0)\). So, \(AB\parallel CD\) (same slope, slope of \(AB\) is \(0\), slope of \(CD\) is \(0\)) and \(|AB| = 3\), \(|CD| = 3\), so \(AB = CD\).
• Check \(AD\) and \(BC\): The vector \(\overrightarrow{AD}=(1 - 1,-1 - 2)=(0,-3)\) and \(\overrightarrow{BC}=(4 - 4,-1 - 2)=(0,-3)\). The slope of \(AD\) is undefined (vertical line) and the slope of \(BC\) is undefined (vertical line), so \(AD\parallel BC\). Also, \(|AD|=|2-(-1)| = 3\) and \(|BC|=|2-(-1)| = 3\), so \(AD = BC\). And the angles between adjacent sides are \(90^{\circ}\) (since \(AB\) is horizontal and \(AD\) is vertical).

Answer:

The figure formed by points \(A\), \(B\), and \(C\) is a right - angled triangle.

Sub - question 2