Question
Use the distance formula to calculate which set of coordinate points make a rectangle.
Answers
GPT-4o mini
To determine whether a set of coordinate points forms a rectangle using the distance formula, we need to follow these steps:
1. **Identify the Points**: Let's assume we have four points \( A(x_1, y_1) \), \( B(x_2, y_2) \), \( C(x_3, y_3) \), and \( D(x_4, y_4) \).
2. **Distance Formula**: The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
3. **Calculate the Distances**: We will calculate the distances between all pairs of points (for example, \( AB \), \( BC \), \( CD \), and \( DA \)). We will also calculate the diagonals \( AC \) and \( BD \).
4. **Check for Rectangles**: In a rectangle:
- The opposite sides are equal in length, i.e., \( AB = CD \) and \( BC = DA \).
- The diagonals are also equal in length, i.e., \( AC = BD \).
### Example Calculation:
Let's say we have the points:
- \( A(0, 0) \)
- \( B(4, 0) \)
- \( C(4, 3) \)
- \( D(0, 3) \)
**Step 1: Calculate the lengths of the sides and diagonals:**
- \( AB \):
\[
AB = \sqrt{(4 - 0)^2 + (0 - 0)^2} = \sqrt{4^2} = 4
\]
- \( BC \):
\[
BC = \sqrt{(4 - 4)^2 + (3 - 0)^2} = \sqrt{3^2} = 3
\]
- \( CD \):
\[
CD = \sqrt{(0 - 4)^2 + (3 - 3)^2} = \sqrt{(-4)^2} = 4
\]
- \( DA \):
\[
DA = \sqrt{(0 - 0)^2 + (0 - 3)^2} = \sqrt{(-3)^2} = 3
\]
- \( AC \):
\[
AC = \sqrt{(4 - 0)^2 + (3 - 0)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5
\]
- \( BD \):
\[
BD = \sqrt{(0 - 4)^2 + (3 - 0)^2} = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = 5
\]
**Step 2: Check the conditions for rectangle:**
- Opposite sides: \( AB = CD (4 = 4) \) and \( BC = DA (3 = 3) \) — both conditions are satisfied.
- Diagonal lengths: \( AC = BD (5 = 5) \) — this condition is also satisfied.
### Conclusion:
From the above calculations, the points \( A(0,0) \), \( B(4,0) \), \( C(4,3) \), and \( D(0,3) \) do indeed form a rectangle.
If you have specific sets of points, please provide them for analysis!
1. **Identify the Points**: Let's assume we have four points \( A(x_1, y_1) \), \( B(x_2, y_2) \), \( C(x_3, y_3) \), and \( D(x_4, y_4) \).
2. **Distance Formula**: The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
3. **Calculate the Distances**: We will calculate the distances between all pairs of points (for example, \( AB \), \( BC \), \( CD \), and \( DA \)). We will also calculate the diagonals \( AC \) and \( BD \).
4. **Check for Rectangles**: In a rectangle:
- The opposite sides are equal in length, i.e., \( AB = CD \) and \( BC = DA \).
- The diagonals are also equal in length, i.e., \( AC = BD \).
### Example Calculation:
Let's say we have the points:
- \( A(0, 0) \)
- \( B(4, 0) \)
- \( C(4, 3) \)
- \( D(0, 3) \)
**Step 1: Calculate the lengths of the sides and diagonals:**
- \( AB \):
\[
AB = \sqrt{(4 - 0)^2 + (0 - 0)^2} = \sqrt{4^2} = 4
\]
- \( BC \):
\[
BC = \sqrt{(4 - 4)^2 + (3 - 0)^2} = \sqrt{3^2} = 3
\]
- \( CD \):
\[
CD = \sqrt{(0 - 4)^2 + (3 - 3)^2} = \sqrt{(-4)^2} = 4
\]
- \( DA \):
\[
DA = \sqrt{(0 - 0)^2 + (0 - 3)^2} = \sqrt{(-3)^2} = 3
\]
- \( AC \):
\[
AC = \sqrt{(4 - 0)^2 + (3 - 0)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5
\]
- \( BD \):
\[
BD = \sqrt{(0 - 4)^2 + (3 - 0)^2} = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = 5
\]
**Step 2: Check the conditions for rectangle:**
- Opposite sides: \( AB = CD (4 = 4) \) and \( BC = DA (3 = 3) \) — both conditions are satisfied.
- Diagonal lengths: \( AC = BD (5 = 5) \) — this condition is also satisfied.
### Conclusion:
From the above calculations, the points \( A(0,0) \), \( B(4,0) \), \( C(4,3) \), and \( D(0,3) \) do indeed form a rectangle.
If you have specific sets of points, please provide them for analysis!