Question

Use the distance formula to calculate which set of coordinate points make a rectangle. (1 point)
A (2,2), B(1,10); C(2,10), D(6,2)
A (6,4), B(4,6); C(6,10), D(4,10)
A (4,2), B(1,10); C(6,10), D(6,4)
A (1,2), B(1,10) ; C(6,10), D(6,2)

Answer 1

To determine which set of points forms a rectangle, we will calculate the distances between pairs of points and check if the distances indicate that the figure has right angles and equal opposite sides.

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

We need to find the lengths of the sides and check for perpendicular intersections (to ensure we have right angles), which can be confirmed by checking that the products of the slopes of the adjacent sides are \(-1\).

Let's analyze each set of points:

### Set 1: A (2,2), B(1,10), C(2,10), D(6,2)

1. **AB**: Distance from A to B
\[
d_{AB} = \sqrt{(1 - 2)^2 + (10 - 2)^2} = \sqrt{1 + 64} = \sqrt{65}
\]

2. **BC**: Distance from B to C
\[
d_{BC} = \sqrt{(2 - 1)^2 + (10 - 10)^2} = \sqrt{1} = 1
\]

3. **CD**: Distance from C to D
\[
d_{CD} = \sqrt{(6 - 2)^2 + (2 - 10)^2} = \sqrt{16 + 64} = \sqrt{80}
\]

4. **DA**: Distance from D to A
\[
d_{DA} = \sqrt{(2 - 6)^2 + (2 - 2)^2} = \sqrt{16} = 4
\]

### Set 2: A (6,4), B(4,6), C(6,10), D(4,10)

1. **AB**:
\[
d_{AB} = \sqrt{(4 - 6)^2 + (6 - 4)^2} = \sqrt{4 + 4} = \sqrt{8}
\]

2. **BC**:
\[
d_{BC} = \sqrt{(6 - 4)^2 + (10 - 6)^2} = \sqrt{4 + 16} = \sqrt{20}
\]

3. **CD**:
\[
d_{CD} = \sqrt{(4 - 6)^2 + (10 - 10)^2} = \sqrt{4} = 2
\]

4. **DA**:
\[
d_{DA} = \sqrt{(6 - 4)^2 + (4 - 10)^2} = \sqrt{4 + 36} = \sqrt{40}
\]

### Set 3: A (4,2), B(1,10), C(6,10), D(6,4)

1. **AB**:
\[
d_{AB} = \sqrt{(1 - 4)^2 + (10 - 2)^2} = \sqrt{9 + 64} = \sqrt{73}
\]

2. **BC**:
\[
d_{BC} = \sqrt{(6 - 1)^2 + (10 - 10)^2} = \sqrt{25} = 5
\]

3. **CD**:
\[
d_{CD} = \sqrt{(6 - 6)^2 + (4 - 10)^2} = \sqrt{36} = 6
\]

4. **DA**:
\[
d_{DA} = \sqrt{(4 - 6)^2 + (2 - 4)^2} = \sqrt{4 + 4} = \sqrt{8}
\]

### Set 4: A (1,2), B(1,10), C(6,10), D(6,2)

1. **AB** (vertical line):
\[
d_{AB} = \sqrt{(1 - 1)^2 + (10 - 2)^2} = \sqrt{64} = 8
\]

2. **BC** (horizontal line):
\[
d_{BC} = \sqrt{(6 - 1)^2 + (10 - 10)^2} = \sqrt{25} = 5
\]

3. **CD** (vertical line):
\[
d_{CD} = \sqrt{(6 - 6)^2 + (2 - 10)^2} = \sqrt{64} = 8
\]

4. **DA** (horizontal line):
\[
d_{DA} = \sqrt{(1 - 6)^2 + (2 - 2)^2} = \sqrt{25} = 5
\]

### Conclusion:
After inspecting the distances for all sets, the distances in Set 4 allow for opposite sides to be equal and parallel (AB = CD and BC = DA). Therefore, **the set that makes a rectangle is:**
- **Set 4**: A (1,2), B(1,10), C(6,10), D(6,2).