Final answer:
The area of the square with a vertex at (7,4) is found using the Pythagorean theorem to determine the length of the diagonal, and then calculating the side length from the diagonal. The area is then the side length squared, which is 32.5 square units.
Step-by-step explanation:
To find the area of the square with a vertex at the point (7,4), we must first determine the side length of the square. Given the square's position in the first quadrant, with two vertices on the axes, the distance from the origin to the vertex at (7,4) will give us the length of the diagonal of the square.
Since the x-coordinate of the vertex is 7, and the y-coordinate is 4, we can use the Pythagorean theorem to find the length of the diagonal: diagonal = √(x² + y²) = √(7² + 4²) = √(49 + 16) = √65.
The side length of the square (s) can be found from the diagonal (d) using the relationship d = s√2. Solving for the side length gives s = d/√2. Therefore, s = √65/√2 = √32.5.
Finally, the area of the square is equal to the side length squared: Area = s². Hence, the area of the square is √32.5² = 32.5 square units.