Question
Use the distance formula to calculate which set of coordinate points make a rectangle. (1 point)
Responses
A (6,4), B(4,6); C(6,10), D(4,10)
A left paren 6 comma 4 right paren comma B right paren 4 comma 6 right paren semi C left paren 6 comma 10 right paren comma D left paren 4 comma 10 right paren
A (2,2), B(1,10); C(2,10), D(6,2)
A left paren 2 comma 2 right paren comma B right paren 1 comma 10 right paren semi C left paren 2 comma 10 right paren comma D left paren 6 comma 2 right paren
A (4,2), B(1,10); C(6,10), D(6,4)
A left paren 4 comma 2 right paren comma B right paren 1 comma 10 right paren semi C left paren 6 comma 10 right paren comma D left paren 6 comma 4 right paren
A (1,2), B(1,10) ; C(6,10), D(6,2)
Responses
A (6,4), B(4,6); C(6,10), D(4,10)
A left paren 6 comma 4 right paren comma B right paren 4 comma 6 right paren semi C left paren 6 comma 10 right paren comma D left paren 4 comma 10 right paren
A (2,2), B(1,10); C(2,10), D(6,2)
A left paren 2 comma 2 right paren comma B right paren 1 comma 10 right paren semi C left paren 2 comma 10 right paren comma D left paren 6 comma 2 right paren
A (4,2), B(1,10); C(6,10), D(6,4)
A left paren 4 comma 2 right paren comma B right paren 1 comma 10 right paren semi C left paren 6 comma 10 right paren comma D left paren 6 comma 4 right paren
A (1,2), B(1,10) ; C(6,10), D(6,2)
Answers
GPT-4o mini
To determine which set of coordinate points forms a rectangle, we need to check if the opposite sides are equal and the adjacent sides are perpendicular. A rectangle has the following properties:
1. Opposite sides are equal in length.
2. Adjacent sides are perpendicular.
We will use the distance formula to find the lengths of the sides. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Let's evaluate each of the given coordinate sets.
### Set 1: A(6,4), B(4,6); C(6,10), D(4,10)
- AB: \(\sqrt{(4 - 6)^2 + (6 - 4)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\)
- BC: \(\sqrt{(6 - 4)^2 + (10 - 6)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}\)
- CD: \(\sqrt{(4 - 6)^2 + (10 - 10)^2} = \sqrt{4 + 0} = 2\)
- DA: \(\sqrt{(6 - 4)^2 + (4 - 10)^2} = \sqrt{4 + 36} = \sqrt{40} = 6.32\)
Not a rectangle since sides are not equal.
### Set 2: A(2,2), B(1,10); C(2,10), D(6,2)
- AB: \(\sqrt{(1 - 2)^2 + (10 - 2)^2} = \sqrt{1 + 64} = \sqrt{65}\)
- BC: \(\sqrt{(2 - 1)^2 + (10 - 10)^2} = \sqrt{1} = 1\)
- CD: \(\sqrt{(6 - 2)^2 + (10 - 2)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}\)
- DA: \(\sqrt{(2 - 6)^2 + (2 - 2)^2} = \sqrt{16} = 4\)
Not a rectangle since sides are not equal.
### Set 3: A(4,2), B(1,10); C(6,10), D(6,4)
- AB: \(\sqrt{(1 - 4)^2 + (10 - 2)^2} = \sqrt{9 + 64} = \sqrt{73}\)
- BC: \(\sqrt{(6 - 1)^2 + (10 - 10)^2} = \sqrt{25} = 5\)
- CD: \(\sqrt{(6 - 6)^2 + (4 - 10)^2} = \sqrt{0 + 36} = 6\)
- DA: \(\sqrt{(4 - 6)^2 + (2 - 4)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\)
Not a rectangle since sides are not equal.
### Set 4: A(1,2), B(1,10); C(6,10), D(6,2)
- AB: \(\sqrt{(1 - 1)^2 + (10 - 2)^2} = \sqrt{0 + 64} = 8\)
- BC: \(\sqrt{(6 - 1)^2 + (10 - 10)^2} = \sqrt{25} = 5\)
- CD: \(\sqrt{(6 - 6)^2 + (2 - 10)^2} = \sqrt{0 + 64} = 8\)
- DA: \(\sqrt{(1 - 6)^2 + (2 - 2)^2} = \sqrt{25} = 5\)
This set forms a rectangle since:
- Opposite sides AB and CD are both \(8\)
- Opposite sides BC and DA are both \(5\)
### Conclusion
**The coordinate points A(1,2), B(1,10); C(6,10), D(6,2) form a rectangle.**
1. Opposite sides are equal in length.
2. Adjacent sides are perpendicular.
We will use the distance formula to find the lengths of the sides. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Let's evaluate each of the given coordinate sets.
### Set 1: A(6,4), B(4,6); C(6,10), D(4,10)
- AB: \(\sqrt{(4 - 6)^2 + (6 - 4)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\)
- BC: \(\sqrt{(6 - 4)^2 + (10 - 6)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}\)
- CD: \(\sqrt{(4 - 6)^2 + (10 - 10)^2} = \sqrt{4 + 0} = 2\)
- DA: \(\sqrt{(6 - 4)^2 + (4 - 10)^2} = \sqrt{4 + 36} = \sqrt{40} = 6.32\)
Not a rectangle since sides are not equal.
### Set 2: A(2,2), B(1,10); C(2,10), D(6,2)
- AB: \(\sqrt{(1 - 2)^2 + (10 - 2)^2} = \sqrt{1 + 64} = \sqrt{65}\)
- BC: \(\sqrt{(2 - 1)^2 + (10 - 10)^2} = \sqrt{1} = 1\)
- CD: \(\sqrt{(6 - 2)^2 + (10 - 2)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}\)
- DA: \(\sqrt{(2 - 6)^2 + (2 - 2)^2} = \sqrt{16} = 4\)
Not a rectangle since sides are not equal.
### Set 3: A(4,2), B(1,10); C(6,10), D(6,4)
- AB: \(\sqrt{(1 - 4)^2 + (10 - 2)^2} = \sqrt{9 + 64} = \sqrt{73}\)
- BC: \(\sqrt{(6 - 1)^2 + (10 - 10)^2} = \sqrt{25} = 5\)
- CD: \(\sqrt{(6 - 6)^2 + (4 - 10)^2} = \sqrt{0 + 36} = 6\)
- DA: \(\sqrt{(4 - 6)^2 + (2 - 4)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\)
Not a rectangle since sides are not equal.
### Set 4: A(1,2), B(1,10); C(6,10), D(6,2)
- AB: \(\sqrt{(1 - 1)^2 + (10 - 2)^2} = \sqrt{0 + 64} = 8\)
- BC: \(\sqrt{(6 - 1)^2 + (10 - 10)^2} = \sqrt{25} = 5\)
- CD: \(\sqrt{(6 - 6)^2 + (2 - 10)^2} = \sqrt{0 + 64} = 8\)
- DA: \(\sqrt{(1 - 6)^2 + (2 - 2)^2} = \sqrt{25} = 5\)
This set forms a rectangle since:
- Opposite sides AB and CD are both \(8\)
- Opposite sides BC and DA are both \(5\)
### Conclusion
**The coordinate points A(1,2), B(1,10); C(6,10), D(6,2) form a rectangle.**