Determining if Four Points Form a Rectangle and Calculating Its Area

Given four points: A(-1, 0), B(6, 7), C(-2, 3), and D(8, 3), we aim to determine if these points form a rectangle. If not, we will calculate the area of the quadrilateral formed by these points using the shoelace formula.

Step-by-Step Solution

Step 1: Calculate the Distances Between the Points

To check if the points form a rectangle, we need to ensure that the distances between each pair of points form two pairs of equal lengths (the sides of the rectangle) and that adjacent sides are perpendicular. We will use the distance formula to calculate the distances:

d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}

Let's calculate the distances:

Distance AB: d_{AB} sqrt{(6 - 0)^2 (7 - (-1))^2} sqrt{6^2 8^2} sqrt{36 64} sqrt{100} 10 Distance AC: d_{AC} sqrt{(-2 - 0)^2 (3 - (-1))^2} sqrt{(-2)^2 4^2} sqrt{4 16} sqrt{20} 2sqrt{5} Distance AD: d_{AD} sqrt{(8 - 0)^2 (3 - (-1))^2} sqrt{8^2 4^2} sqrt{64 16} sqrt{80} 4sqrt{5} Distance BC: d_{BC} sqrt{(-2 - 6)^2 (3 - 7)^2} sqrt{(-8)^2 (-4)^2} sqrt{64 16} sqrt{80} 4sqrt{5} Distance BD: d_{BD} sqrt{(8 - 6)^2 (3 - 7)^2} sqrt{2^2 (-4)^2} sqrt{4 16} sqrt{20} 2sqrt{5} Distance CD: d_{CD} sqrt{(8 - (-2))^2 (3 - 3)^2} sqrt{10^2} 10

Summary of Distances:

d_{AB} 10 d_{AC} 2sqrt{5} d_{AD} 4sqrt{5} d_{BC} 4sqrt{5} d_{BD} 2sqrt{5} d_{CD} 10

We can see that d_{AB} d_{CD} 10 and d_{AD} d_{BC} 4sqrt{5}. This indicates that the points do not necessarily form a rectangle but do have two pairs of equal lengths.

Step 2: Check the Slopes

We now check the slopes of the sides to ensure they are perpendicular:

Slope of AB: m_{AB} frac{7 - (-1)}{6 - 0} frac{8}{6} frac{4}{3} Slope of BC: m_{BC} frac{3 - 7}{-2 - 6} frac{-4}{-8} frac{1}{2} Slope of CD: m_{CD} frac{3 - 3}{8 - (-2)} 0 (horizontal line) Slope of DA: m_{DA} frac{3 - (-1)}{8 - 0} frac{4}{8} frac{1}{2}

To check if two lines are perpendicular, we multiply their slopes and ensure the product is -1:

m_{AB} cdot m_{BC} frac{4}{3} cdot frac{1}{2} frac{2}{3}; not perpendicular m_{AB} cdot m_{DA} frac{4}{3} cdot frac{1}{2} frac{2}{3}; not perpendicular m_{CD} cdot m_{DA} 0; not perpendicular (horizontal and vertical lines)

Since the slopes do not indicate perpendicularity between adjacent sides, the points do not form a rectangle. However, the distances indicate pairs of equal lengths.

Area of the Quadrilateral Formed

Even though the points do not form a rectangle, we can still calculate the area of the quadrilateral using the shoelace formula:

text{Area} frac{1}{2} left( sum_{i1}^{n-1} (x_i y_{i 1} - y_i x_{i 1}) (x_n y_1 - y_n x_1) right)

Substituting in the coordinates:

text{Area} frac{1}{2} left( 0 cdot 7 6 cdot 3 (-2) cdot 3 8 cdot (-1) - (-1) cdot 6 7 cdot (-2) 3 cdot 8 3 cdot 0 right)

Calculating:

text{Area} frac{1}{2} left( 0 18 - 6 - 8 6 - 14 24 0 right)

frac{1}{2} left( 4 - (-14) right) frac{1}{2} left( 4 14 right) frac{1}{2} times 18 9

Final Result: The area of the quadrilateral formed by the points is 9 square units.