Question

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

Answer 1

To determine which set of coordinate points forms a rectangle, we will use the distance formula between each pair of points and check for the properties of a rectangle:

1. Opposite sides must be equal in length.
2. Adjacent sides must be perpendicular.

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

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

Now let's calculate the distances for each set of coordinates.

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

1. **AB**:
\[
AB = \sqrt{(1 - 1)^2 + (10 - 2)^2} = \sqrt{0 + 64} = \sqrt{64} = 8
\]

2. **BC**:
\[
BC = \sqrt{(6 - 1)^2 + (10 - 10)^2} = \sqrt{25 + 0} = \sqrt{25} = 5
\]

3. **CD**:
\[
CD = \sqrt{(6 - 6)^2 + (10 - 2)^2} = \sqrt{0 + 64} = \sqrt{64} = 8
\]

4. **DA**:
\[
DA = \sqrt{(1 - 6)^2 + (2 - 2)^2} = \sqrt{25 + 0} = \sqrt{25} = 5
\]

Lengths are: \(AB = CD = 8\) and \(BC = DA = 5\); opposite sides are equal.

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

1. **AB**:
\[
AB = \sqrt{(4 - 6)^2 + (6 - 4)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]

2. **BC**:
\[
BC = \sqrt{(6 - 4)^2 + (10 - 6)^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.47
\]

3. **CD**:
\[
CD = \sqrt{(4 - 6)^2 + (10 - 10)^2} = \sqrt{4 + 0} = \sqrt{4} = 2
\]

4. **DA**:
\[
DA = \sqrt{(6 - 4)^2 + (4 - 10)^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.32
\]

Not all sides match; it does not form a rectangle.

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

1. **AB**:
\[
AB = \sqrt{(1 - 2)^2 + (10 - 2)^2} = \sqrt{1 + 64} = \sqrt{65}
\]

2. **BC**:
\[
BC = \sqrt{(2 - 1)^2 + (10 - 10)^2} = \sqrt{1 + 0} = 1
\]

3. **CD**:
\[
CD = \sqrt{(6 - 2)^2 + (2 - 10)^2} = \sqrt{16 + 64} = \sqrt{80}
\]

4. **DA**:
\[
DA = \sqrt{(2 - 6)^2 + (2 - 2)^2} = \sqrt{16 + 0} = 4
\]

Not all sides match; it does not form a rectangle.

### Conclusion
Only **Option A** with points A(1, 2), B(1, 10), C(6, 10), and D(6, 2) forms a rectangle, as both pairs of opposite sides are equal in length.

**The answer is Option A.**