Question
Use the distance formula to calculate which set of coordinate points make a rectangle. (1 point)
Responses
A (1,2), B(1,10) ; C(6,10), D(6,2)
A (1,2), B(1,10) ; C(6,10), D(6,2)
- no response given
A (6,4), B(4,6); C(6,10), D(4,10)
A (6,4), B(4,6); C(6,10), D(4,10)
- no response given
A (4,2), B(1,10); C(6,10), D(6,4)
A (4,2), B(1,10); C(6,10), D(6,4)
- no response given
A (2,2), B(1,10); C(2,10), D(6,2)
Responses
A (1,2), B(1,10) ; C(6,10), D(6,2)
A (1,2), B(1,10) ; C(6,10), D(6,2)
- no response given
A (6,4), B(4,6); C(6,10), D(4,10)
A (6,4), B(4,6); C(6,10), D(4,10)
- no response given
A (4,2), B(1,10); C(6,10), D(6,4)
A (4,2), B(1,10); C(6,10), D(6,4)
- no response given
A (2,2), B(1,10); C(2,10), D(6,2)
Answers
Answered by
GPT-4o mini
To determine which set of coordinate points forms a rectangle, we will use the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
A rectangle has opposite sides that are equal in length and all angles are right angles. We will check each set of points to see if the distances between the points confirm that they form a rectangle.
1. **Set 1: A (1,2), B(1,10), C(6,10), D(6,2)**
- AB: Distance = \(\sqrt{(1-1)^2 + (10-2)^2} = \sqrt{0 + 64} = 8\)
- BC: Distance = \(\sqrt{(6-1)^2 + (10-10)^2} = \sqrt{25 + 0} = 5\)
- CD: Distance = \(\sqrt{(6-6)^2 + (10-2)^2} = \sqrt{0 + 64} = 8\)
- DA: Distance = \(\sqrt{(1-6)^2 + (2-2)^2} = \sqrt{25 + 0} = 5\)
- Result: Opposite sides are equal (8 and 5).
2. **Set 2: A (6,4), B(4,6), C(6,10), D(4,10)**
- AB: Distance = \(\sqrt{(4-6)^2 + (6-4)^2} = \sqrt{4 + 4} = \sqrt{8}\)
- BC: Distance = \(\sqrt{(6-4)^2 + (10-6)^2} = \sqrt{4 + 16} = \sqrt{20}\)
- CD: Distance = \(\sqrt{(4-6)^2 + (10-10)^2} = \sqrt{4 + 0} = 2\)
- DA: Distance = \(\sqrt{(6-4)^2 + (4-10)^2} = \sqrt{4 + 36} = \sqrt{40}\)
- Result: Does not have equal opposite sides.
3. **Set 3: A (4,2), B(1,10), C(6,10), D(6,4)**
- AB: Distance = \(\sqrt{(1-4)^2 + (10-2)^2} = \sqrt{9 + 64} = \sqrt{73}\)
- BC: Distance = \(\sqrt{(6-1)^2 + (10-10)^2} = \sqrt{25 + 0} = 5\)
- CD: Distance = \(\sqrt{(6-6)^2 + (10-4)^2} = \sqrt{0 + 36} = 6\)
- DA: Distance = \(\sqrt{(4-6)^2 + (2-4)^2} = \sqrt{4 + 4} = 2\)
- Result: Does not have equal opposite sides.
4. **Set 4: A (2,2), B(1,10), C(2,10), D(6,2)**
- AB: Distance = \(\sqrt{(1-2)^2 + (10-2)^2} = \sqrt{1 + 64} = \sqrt{65}\)
- BC: Distance = \(\sqrt{(2-1)^2 + (10-10)^2} = \sqrt{1 + 0} = 1\)
- CD: Distance = \(\sqrt{(6-2)^2 + (2-10)^2} = \sqrt{16 + 64} = \sqrt{80}\)
- DA: Distance = \(\sqrt{(2-6)^2 + (2-2)^2} = \sqrt{16 + 0} = 4\)
- Result: Does not have equal opposite sides.
**Final Verdict:**
Among the options presented, **Set 1: A (1,2), B(1,10), C(6,10), D(6,2)** is the only set of coordinates that forms a rectangle because it satisfies the conditions of having equal opposite sides.
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
A rectangle has opposite sides that are equal in length and all angles are right angles. We will check each set of points to see if the distances between the points confirm that they form a rectangle.
1. **Set 1: A (1,2), B(1,10), C(6,10), D(6,2)**
- AB: Distance = \(\sqrt{(1-1)^2 + (10-2)^2} = \sqrt{0 + 64} = 8\)
- BC: Distance = \(\sqrt{(6-1)^2 + (10-10)^2} = \sqrt{25 + 0} = 5\)
- CD: Distance = \(\sqrt{(6-6)^2 + (10-2)^2} = \sqrt{0 + 64} = 8\)
- DA: Distance = \(\sqrt{(1-6)^2 + (2-2)^2} = \sqrt{25 + 0} = 5\)
- Result: Opposite sides are equal (8 and 5).
2. **Set 2: A (6,4), B(4,6), C(6,10), D(4,10)**
- AB: Distance = \(\sqrt{(4-6)^2 + (6-4)^2} = \sqrt{4 + 4} = \sqrt{8}\)
- BC: Distance = \(\sqrt{(6-4)^2 + (10-6)^2} = \sqrt{4 + 16} = \sqrt{20}\)
- CD: Distance = \(\sqrt{(4-6)^2 + (10-10)^2} = \sqrt{4 + 0} = 2\)
- DA: Distance = \(\sqrt{(6-4)^2 + (4-10)^2} = \sqrt{4 + 36} = \sqrt{40}\)
- Result: Does not have equal opposite sides.
3. **Set 3: A (4,2), B(1,10), C(6,10), D(6,4)**
- AB: Distance = \(\sqrt{(1-4)^2 + (10-2)^2} = \sqrt{9 + 64} = \sqrt{73}\)
- BC: Distance = \(\sqrt{(6-1)^2 + (10-10)^2} = \sqrt{25 + 0} = 5\)
- CD: Distance = \(\sqrt{(6-6)^2 + (10-4)^2} = \sqrt{0 + 36} = 6\)
- DA: Distance = \(\sqrt{(4-6)^2 + (2-4)^2} = \sqrt{4 + 4} = 2\)
- Result: Does not have equal opposite sides.
4. **Set 4: A (2,2), B(1,10), C(2,10), D(6,2)**
- AB: Distance = \(\sqrt{(1-2)^2 + (10-2)^2} = \sqrt{1 + 64} = \sqrt{65}\)
- BC: Distance = \(\sqrt{(2-1)^2 + (10-10)^2} = \sqrt{1 + 0} = 1\)
- CD: Distance = \(\sqrt{(6-2)^2 + (2-10)^2} = \sqrt{16 + 64} = \sqrt{80}\)
- DA: Distance = \(\sqrt{(2-6)^2 + (2-2)^2} = \sqrt{16 + 0} = 4\)
- Result: Does not have equal opposite sides.
**Final Verdict:**
Among the options presented, **Set 1: A (1,2), B(1,10), C(6,10), D(6,2)** is the only set of coordinates that forms a rectangle because it satisfies the conditions of having equal opposite sides.