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Sagot :
Let's solve the problem step-by-step:
1. Finding the Coordinates of B and D:
Given:
- A(-1, -1) and C(1, 1) are the coordinates of two opposite vertices of the square.
### Step-by-Step Solution:
#### Step 1: Calculate the Center of the Square
The center of the square (midpoint of AC) can be calculated as:
[tex]\[ \text{center} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
[tex]\[ \text{center} = \left( \frac{-1 + 1}{2}, \frac{-1 + 1}{2} \right) = (0, 0) \][/tex]
So, the center of the square is at (0, 0).
#### Step 2: Calculate the Side Length of the Square
The distance between A and C can be calculated using the distance formula:
[tex]\[ AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
[tex]\[ AC = \sqrt{(1 - (-1))^2 + (1 - (-1))^2} = \sqrt{(1 + 1)^2 + (1 + 1)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \][/tex]
Since AC is the diagonal of the square and the diagonal of a square is [tex]\( \sqrt{2} \)[/tex] times the side length, the side length [tex]\( s \)[/tex] is:
[tex]\[ s = \frac{AC}{\sqrt{2}} = \frac{2\sqrt{2}}{\sqrt{2}} = 2 \][/tex]
#### Step 3: Determine the Coordinates of B and D
We know that B and D are aligned perpendicularly along with the square's sides from A and C. Given that A and C are on a diagonal, B and D would lie symmetrically on the other diagonal.
By symmetry and distance considerations, from the center (0, 0), B and D will also be equidistant from the center along the axes perpendicular to A and C.
Using the properties of the square, the coordinates of B and D can be calculated using a rotation by 90 degrees around the center of the square from point A:
1. Rotate A(-1, -1) 90 degrees counterclockwise around the center (0, 0):
[tex]\[ B(x, y) = (-(-1), -1) = (1, -1) \][/tex]
2. Rotate C(1, 1) 90 degrees counterclockwise around the center (0, 0):
[tex]\[ D(x, y) = (-1, 1) So, the coordinates of B (1, -1) and D (-1, 1). 2. Finding Equations of the Sides: #### Step 1: Equation of AB: Points A(-1, -1) and B(1, -1): It is a horizontal line: \[ y = -1 \][/tex]
#### Step 2: Equation of BC:
Points B(1, -1) and C(1, 1):
It is a vertical line:
[tex]\[ x = 1 \][/tex]
#### Step 3: Equation of CD:
Points C(1, 1) and D(-1, 1):
It is a horizontal line:
[tex]\[ y = 1 \][/tex]
#### Step 4: Equation of DA:
Points D(-1, 1) and A(-1, -1):
It is a vertical line:
[tex]\[ x = -1 \][/tex]
Summary of Results:
- Coordinates of B: (1, -1)
- Coordinates of D: (-1, 1)
- Equations of the sides:
- AB: [tex]\( y = -1 \)[/tex]
- BC: [tex]\( x = 1 \)[/tex]
- CD: [tex]\( y = 1 \)[/tex]
- DA: [tex]\( x = -1 \)[/tex]
1. Finding the Coordinates of B and D:
Given:
- A(-1, -1) and C(1, 1) are the coordinates of two opposite vertices of the square.
### Step-by-Step Solution:
#### Step 1: Calculate the Center of the Square
The center of the square (midpoint of AC) can be calculated as:
[tex]\[ \text{center} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
[tex]\[ \text{center} = \left( \frac{-1 + 1}{2}, \frac{-1 + 1}{2} \right) = (0, 0) \][/tex]
So, the center of the square is at (0, 0).
#### Step 2: Calculate the Side Length of the Square
The distance between A and C can be calculated using the distance formula:
[tex]\[ AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
[tex]\[ AC = \sqrt{(1 - (-1))^2 + (1 - (-1))^2} = \sqrt{(1 + 1)^2 + (1 + 1)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \][/tex]
Since AC is the diagonal of the square and the diagonal of a square is [tex]\( \sqrt{2} \)[/tex] times the side length, the side length [tex]\( s \)[/tex] is:
[tex]\[ s = \frac{AC}{\sqrt{2}} = \frac{2\sqrt{2}}{\sqrt{2}} = 2 \][/tex]
#### Step 3: Determine the Coordinates of B and D
We know that B and D are aligned perpendicularly along with the square's sides from A and C. Given that A and C are on a diagonal, B and D would lie symmetrically on the other diagonal.
By symmetry and distance considerations, from the center (0, 0), B and D will also be equidistant from the center along the axes perpendicular to A and C.
Using the properties of the square, the coordinates of B and D can be calculated using a rotation by 90 degrees around the center of the square from point A:
1. Rotate A(-1, -1) 90 degrees counterclockwise around the center (0, 0):
[tex]\[ B(x, y) = (-(-1), -1) = (1, -1) \][/tex]
2. Rotate C(1, 1) 90 degrees counterclockwise around the center (0, 0):
[tex]\[ D(x, y) = (-1, 1) So, the coordinates of B (1, -1) and D (-1, 1). 2. Finding Equations of the Sides: #### Step 1: Equation of AB: Points A(-1, -1) and B(1, -1): It is a horizontal line: \[ y = -1 \][/tex]
#### Step 2: Equation of BC:
Points B(1, -1) and C(1, 1):
It is a vertical line:
[tex]\[ x = 1 \][/tex]
#### Step 3: Equation of CD:
Points C(1, 1) and D(-1, 1):
It is a horizontal line:
[tex]\[ y = 1 \][/tex]
#### Step 4: Equation of DA:
Points D(-1, 1) and A(-1, -1):
It is a vertical line:
[tex]\[ x = -1 \][/tex]
Summary of Results:
- Coordinates of B: (1, -1)
- Coordinates of D: (-1, 1)
- Equations of the sides:
- AB: [tex]\( y = -1 \)[/tex]
- BC: [tex]\( x = 1 \)[/tex]
- CD: [tex]\( y = 1 \)[/tex]
- DA: [tex]\( x = -1 \)[/tex]
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