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Sagot :
Sure, let's break down the process of determining whether the points [tex]\((2, -2)\)[/tex], [tex]\((8, 4)\)[/tex], [tex]\((5, 7)\)[/tex], and [tex]\((-1, 1)\)[/tex] form the vertices of a rectangle, and how to compute the area if they do.
### Step-by-Step Solution
1. Label the Points:
- [tex]\(A = (2, -2)\)[/tex]
- [tex]\(B = (8, 4)\)[/tex]
- [tex]\(C = (5, 7)\)[/tex]
- [tex]\(D = (-1, 1)\)[/tex]
2. Distance Between Points:
We need to calculate the distances between each pair of points to check whether opposite sides are equal (which is a necessary criterion for a rectangle).
- Distance AB:
[tex]\[ AB = \sqrt{(8 - 2)^2 + (4 + 2)^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} \approx 8.49 \][/tex]
- Distance BC:
[tex]\[ BC = \sqrt{(5 - 8)^2 + (7 - 4)^2} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24 \][/tex]
- Distance CD:
[tex]\[ CD = \sqrt{(5 - (-1))^2 + (7 - 1)^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} \approx 8.49 \][/tex]
- Distance DA:
[tex]\[ DA = \sqrt{(2 - (-1))^2 + (-2 - 1)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24 \][/tex]
- Distance AC:
[tex]\[ AC = \sqrt{(5 - 2)^2 + (7 + 2)^2} = \sqrt{3^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} \approx 9.49 \][/tex]
- Distance BD:
[tex]\[ BD = \sqrt{(8 - (-1))^2 + (4 - 1)^2} = \sqrt{9^2 + 3^2} = \sqrt{81 + 9} = \sqrt{90} \approx 9.49 \][/tex]
3. Check Rectangle Condition:
For the given points to form a rectangle:
- Opposite sides must be equal: [tex]\(AB = CD\)[/tex] and [tex]\(BC = DA\)[/tex]
- The diagonals must be equal: [tex]\(AC = BD\)[/tex]
From our calculations:
[tex]\[ AB \approx 8.49, \quad CD \approx 8.49, \quad BC \approx 4.24, \quad DA \approx 4.24, \quad AC \approx 9.49, \quad BD \approx 9.49 \][/tex]
Since [tex]\(AB = CD\)[/tex], [tex]\(BC = DA\)[/tex], and [tex]\(AC = BD\)[/tex], the points do indeed form a rectangle.
4. Calculate the Area:
The area of a rectangle can be calculated using the lengths of two adjacent sides.
[tex]\[ \text{Area} = AB \times BC \approx 8.49 \times 4.24 = 35.99 \][/tex]
Therefore, the points [tex]\((2, -2)\)[/tex], [tex]\((8, 4)\)[/tex], [tex]\((5, 7)\)[/tex], and [tex]\((-1, 1)\)[/tex] form the vertices of a rectangle, and the area of this rectangle is approximately [tex]\(36\)[/tex].
### Understanding Collinearity:
(a) In geometry, collinearity refers to the property of points lying on a single straight line. If three or more points are collinear, it means that there exists a straight line that passes through all the points.
### Step-by-Step Solution
1. Label the Points:
- [tex]\(A = (2, -2)\)[/tex]
- [tex]\(B = (8, 4)\)[/tex]
- [tex]\(C = (5, 7)\)[/tex]
- [tex]\(D = (-1, 1)\)[/tex]
2. Distance Between Points:
We need to calculate the distances between each pair of points to check whether opposite sides are equal (which is a necessary criterion for a rectangle).
- Distance AB:
[tex]\[ AB = \sqrt{(8 - 2)^2 + (4 + 2)^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} \approx 8.49 \][/tex]
- Distance BC:
[tex]\[ BC = \sqrt{(5 - 8)^2 + (7 - 4)^2} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24 \][/tex]
- Distance CD:
[tex]\[ CD = \sqrt{(5 - (-1))^2 + (7 - 1)^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} \approx 8.49 \][/tex]
- Distance DA:
[tex]\[ DA = \sqrt{(2 - (-1))^2 + (-2 - 1)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24 \][/tex]
- Distance AC:
[tex]\[ AC = \sqrt{(5 - 2)^2 + (7 + 2)^2} = \sqrt{3^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} \approx 9.49 \][/tex]
- Distance BD:
[tex]\[ BD = \sqrt{(8 - (-1))^2 + (4 - 1)^2} = \sqrt{9^2 + 3^2} = \sqrt{81 + 9} = \sqrt{90} \approx 9.49 \][/tex]
3. Check Rectangle Condition:
For the given points to form a rectangle:
- Opposite sides must be equal: [tex]\(AB = CD\)[/tex] and [tex]\(BC = DA\)[/tex]
- The diagonals must be equal: [tex]\(AC = BD\)[/tex]
From our calculations:
[tex]\[ AB \approx 8.49, \quad CD \approx 8.49, \quad BC \approx 4.24, \quad DA \approx 4.24, \quad AC \approx 9.49, \quad BD \approx 9.49 \][/tex]
Since [tex]\(AB = CD\)[/tex], [tex]\(BC = DA\)[/tex], and [tex]\(AC = BD\)[/tex], the points do indeed form a rectangle.
4. Calculate the Area:
The area of a rectangle can be calculated using the lengths of two adjacent sides.
[tex]\[ \text{Area} = AB \times BC \approx 8.49 \times 4.24 = 35.99 \][/tex]
Therefore, the points [tex]\((2, -2)\)[/tex], [tex]\((8, 4)\)[/tex], [tex]\((5, 7)\)[/tex], and [tex]\((-1, 1)\)[/tex] form the vertices of a rectangle, and the area of this rectangle is approximately [tex]\(36\)[/tex].
### Understanding Collinearity:
(a) In geometry, collinearity refers to the property of points lying on a single straight line. If three or more points are collinear, it means that there exists a straight line that passes through all the points.
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