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:

Sub - Question 1

Step 1: Analyze coordinates of points

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 \(|4 - 1|=3\) (using the distance formula for horizontal lines \(d=\vert x_2 - x_1\vert\) when \(y_1 = y_2\)).

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 \(|2-(-1)| = 3\) (using the distance formula for vertical lines \(d=\vert y_2 - y_1\vert\) when \(x_1=x_2\)).

Also, the slope of \(AB\) (horizontal line) is \(0\) (since \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - 2}{4 - 1}=0\)) and the slope of \(BC\) (vertical line) is undefined (since \(m=\frac{-1 - 2}{4 - 4}\) is undefined), so \(AB\perp BC\).

Step 2: Determine the shape

Since we have three points where two sides are perpendicular and of length \(3\) (and we connect \(A\) to \(B\) to \(C\) to \(A\)), the figure formed is a right triangle.

Step 1: Identify the base and height of the triangle

In a right triangle, the two legs can be considered as the base and the height. From the coordinates, the length of \(AB\) (horizontal leg) is \(b = 4 - 1=3\) and the length of \(BC\) (vertical leg) is \(h=2-(-1) = 3\).

Step 2: Use the area formula for a right triangle

The formula for the area of a right triangle is \(A=\frac{1}{2}\times\text{base}\times\text{height}\). Substituting \(b = 3\) and \(h = 3\) into the formula, we get \(A=\frac{1}{2}\times3\times3\).

Step 3: Calculate the area

\(\frac{1}{2}\times3\times3=\frac{9}{2}=4.5\)

Step 1: Recall the properties of a rectangle

In a rectangle, opposite sides are equal and parallel, and adjacent sides are perpendicular. For quadrilateral \(ABCD\) to be a rectangle, \(AB\parallel CD\) and \(AB = CD\), \(BC\parallel AD\) and \(BC=AD\).

We know \(A(1,2)\), \(B(4,2)\), \(C(4,-1)\). Since \(AB\) is horizontal (from \(x = 1\) to \(x = 4\) at \(y = 2\)), \(CD\) should also be horizontal. So the \(y\) - coordinate of \(D\) should be the same as the \(y\) - coordinate of \(A\) (because \(BC\) is vertical from \(y = 2\) to \(y=-1\) at \(x = 4\), so \(AD\) should be vertical).

The \(x\) - coordinate of \(D\) should be the same as the \(x\) - coordinate of \(A\) (since \(AD\) is vertical) and the \(y\) - coordinate of \(D\) should be the same as the \(y\) - coordinate of \(C\) (since \(CD\) is horizontal)? Wait, no. Let's use the vector approach.

Vector \(\overrightarrow{AB}=(4 - 1,2 - 2)=(3,0)\). Vector \(\overrightarrow{BC}=(4 - 4,-1 - 2)=(0,-3)\). To find \(D\), we know that \(\overrightarrow{AD}=\overrightarrow{BC}\) (since in a rectangle \(\overrightarrow{AD}=\overrightarrow{BC}\)).

If \(A=(x_A,y_A)=(1,2)\) and \(\overrightarrow{AD}=(0,-3)\) (same as \(\overrightarrow{BC}\)), then \(D=(x_A+0,y_A-3)=(1,2 - 3)=(1,-1)\).

Step 2: Verify the properties

• Check \(AB\) and \(CD\): \(AB\) has endpoints \((1,2)\) and \((4,2)\), \(CD\) has endpoints \((4,-1)\) and \((1,-1)\). The length of \(AB\) is \(3\) ( \(4 - 1\)) and the length of \(CD\) is \(3\) ( \(4 - 1\)). The slope of \(AB\) is \(0\) and the slope of \(CD\) is \(0\), so \(AB\parallel CD\).
• Check \(BC\) and \(AD\): \(BC\) has endpoints \((4,2)\) and \((4,-1)\), \(AD\) has endpoints \((1,2)\) and \((1,-1)\). The length of \(BC\) is \(3\) ( \(2-(-1)\)) and the length of \(AD\) is \(3\) ( \(2-(-1)\)). The slope of \(BC\) is undefined and the slope of \(AD\) is undefined, so \(BC\parallel AD\).
• Check adjacent sides: \(AB\) (slope \(0\)) and \(BC\) (slope undefined) are perpendicular, \(BC\) (slope undefined) and \(CD\) (slope \(0\)) are perpendicular, \(CD\) (slope \(0\)) and \(AD\) (slope undefined) are perpendicular, \(AD\) (slope undefined) and \(AB\) (slope \(0\)) are perpendicular.

Answer:

The figure is a right triangle.

Sub - Question 2