Each side of a square is of length 4. The center of the square is (3,7) and the sides of its diagonals are parallel to y=x. Then, what are the coordinates of the vertices?

To find the coordinates of the vertices of a square with a center at (3, 7) and a side length of 4, we first determine the distance from the center to each vertex. Since the diagonals are parallel to the line y = x, the square is rotated 45 degrees.

Calculating Half the Diagonal Length

The diagonal of a square can be calculated using the formula:

• Diagonal = side length × √2

For our square:

• Diagonal = 4 × √2 ≈ 5.66

Half of the diagonal is:

• Half Diagonal = 5.66 / 2 ≈ 2.83

Finding the Vertex Coordinates

Since the square is rotated, we can find the coordinates of the vertices by adding and subtracting the half diagonal from the center coordinates:

• Vertex 1: (3 + 2.83, 7 + 2.83) = (5.83, 9.83)
• Vertex 2: (3 - 2.83, 7 + 2.83) = (0.17, 9.83)
• Vertex 3: (3 - 2.83, 7 - 2.83) = (0.17, 4.17)
• Vertex 4: (3 + 2.83, 7 - 2.83) = (5.83, 4.17)

Final Vertex Coordinates

The coordinates of the vertices of the square are:

• (5.83, 9.83)
• (0.17, 9.83)
• (0.17, 4.17)
• (5.83, 4.17)

To find the coordinates of the vertices of a square with a given center and side length, we can follow a systematic approach.

Understanding the Square's Properties

The square has a center at (3, 7) and a side length of 4. Since the diagonals are parallel to the line y = x, the square is rotated 45 degrees from the standard position.

Calculating Half the Diagonal Length

The diagonal of a square can be calculated using the formula:

• Diagonal = side length × √2

For our square:

• Diagonal = 4 × √2 ≈ 5.66

Half of the diagonal is:

• Half Diagonal = (4 × √2) / 2 = 2√2 ≈ 2.83

Finding the Vertices

To find the vertices, we can determine their coordinates by moving from the center (3, 7) in the directions of the square's corners. The coordinates of the vertices can be calculated as follows:

• Vertex 1: (3 - 2, 7 + 2) = (1, 9)
• Vertex 2: (3 + 2, 7 + 2) = (5, 9)
• Vertex 3: (3 + 2, 7 - 2) = (5, 5)
• Vertex 4: (3 - 2, 7 - 2) = (1, 5)

Final Coordinates

The coordinates of the vertices of the square are:

• (1, 9)
• (5, 9)
• (5, 5)
• (1, 5)