To solve this problem, we need to determine the coordinates of another vertex of a square given that one vertex is at (-2, 5) and the area of the square is 49 square units.
The area of a square is given by the formula \( \text{Area} = s^2 \), where \( s \) is the length of one side of the square. Therefore, we can find the side length:
\[ s^2 = 49 \implies s = \sqrt{49} = 7 \]
This means each side of the square is 7 units long.
To find other possible vertices, we can consider the square positioned in different orientations: either aligned with the axes or rotated.
1. Aligned with Axes:
If the square is aligned with the axes, the other vertices can be calculated by moving 7 units from the given vertex (-2, 5) in both the x and y directions.
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Moving 7 units right: \((-2 + 7, 5) = (5, 5)\)
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Moving 7 units left: \((-2 - 7, 5) = (-9, 5)\)
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Moving 7 units up: \((-2, 5 + 7) = (-2, 12)\)
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Moving 7 units down: \((-2, 5 - 7) = (-2, -2)\)
So the possible vertices when aligned with axes are:
- (5, 5)
- (-9, 5)
- (-2, 12)
- (-2, -2)
2. Rotated Square:
If the square is rotated, there are more possibilities. A square can be rotated at 45 degrees from its axis-aligned position, creating potential diagonal movements.
- Using the diagonal formula: We can use distance calculations to find coordinates at a 45-degree angle. The other vertices would be at positions that can satisfy the distance from (-2, 5) using geometry.
For instance:
- Moving diagonally \(7 \sqrt{2}\) units (which corresponds to the square’s diagonal or vertices at 45 degrees):
- Calculate the coordinates by using angle rotation or distance measure.
This will yield more possibilities, but will require the use of sine and cosine to properly calculate diagonally.
Final Answer: The potential coordinates of another vertex could include:
- (5, 5)
- (-9, 5)
- (-2, 12)
- (-2, -2)
Additional vertices can be determined based on rotated positions, which would need trigonometric calculations. You would typically find every vertex through an approach that employs vector movement and direction calculations based on this original vertex.