To show that the points [tex]\( A(2, -2) \)[/tex], [tex]\( B(8, 4) \)[/tex], [tex]\( C(5, 7) \)[/tex], and [tex]\( D(-1, 1) \)[/tex] form the vertices of a rectangle, we need to verify that opposite sides are equal in length and that the diagonals are equal.
### Step 1: Calculate the Distances Between Consecutive Vertices
1. Distance [tex]\( AB \)[/tex]:
The distance between points [tex]\( A(2, -2) \)[/tex] and [tex]\( B(8, 4) \)[/tex] is calculated using the distance formula:
[tex]\[
AB = \sqrt{(8 - 2)^2 + (4 - (-2))^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} = 8.48528137423857
\][/tex]
2. Distance [tex]\( BC \)[/tex]:
The distance between points [tex]\( B(8, 4) \)[/tex] and [tex]\( C(5, 7) \)[/tex] is:
[tex]\[
BC = \sqrt{(5 - 8)^2 + (7 - 4)^2} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 4.242640687119285
\][/tex]
3. Distance [tex]\( CD \)[/tex]:
The distance between points [tex]\( C(5, 7) \)[/tex] and [tex]\( D(-1, 1) \)[/tex] is:
[tex]\[
CD = \sqrt{(-1 - 5)^2 + (1 - 7)^2} = \sqrt{(-6)^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} = 8.48528137423857
\][/tex]
4. Distance [tex]\( DA \)[/tex]:
The distance between points [tex]\( D(-1, 1) \)[/tex] and [tex]\( A(2, -2) \)[/tex] is:
[tex]\[
DA = \sqrt{(2 - (-1))^2 + (-2 - 1)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 4.242640687119285
\][/tex]
### Step 2: Calculate the Diagonals
1. Distance [tex]\( AC \)[/tex]:
The distance between points [tex]\( A(2, -2) \)[/tex] and [tex]\( C(5, 7) \)[/tex] is:
[tex]\[
AC = \sqrt{(5 - 2)^2 + (7 - (-2))^2} = \sqrt{3^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} = 9.486832980505138
\][/tex]
2. Distance [tex]\( BD \)[/tex]:
The distance between points [tex]\( B(8, 4) \)[/tex] and [tex]\( D(-1, 1) \)[/tex] is:
[tex]\[
BD = \sqrt{(-1 - 8)^2 + (1 - 4)^2} = \sqrt{(-9)^2 + (-3)^2} = \sqrt{81 + 9} = \sqrt{90} = 9.486832980505138
\][/tex]
### Step 3: Verify the Properties of the Rectangle
We need to check the following conditions:
1. Opposite sides are equal:
[tex]\[
AB = CD = 8.48528137423857 \quad \text{and} \quad BC = DA = 4.242640687119285
\][/tex]
2. Diagonals are equal:
[tex]\[
AC = BD = 9.486832980505138
\][/tex]
Since both conditions are satisfied, the points [tex]\( A(2, -2) \)[/tex], [tex]\( B(8, 4) \)[/tex], [tex]\( C(5, 7) \)[/tex], and [tex]\( D(-1, 1) \)[/tex] form the vertices of a rectangle [tex]\( ABCD \)[/tex].
