To find another vertex of a square given one vertex and the perimeter, we first need to calculate the side length of the square. The perimeter of a square is given by the formula:
\[ \text{Perimeter} = 4 \times \text{side length} \]
Given that the perimeter is 36 units, we can find the side length as follows:
\[ 4s = 36 \implies s = \frac{36}{4} = 9 \]
Thus, the side length of the square is 9 units.
Next, we know that a square has 4 vertices that are equidistant (the side length) from each other at right angles. We can denote the known vertex of the square as \( A(3, 5) \).
To find possible coordinates for another vertex, we need to consider that the other vertices must either be 9 units horizontally or vertically away from \( A(3, 5) \), or they can also be at a 45-degree angle from \( A \).
1. Horizontal/Vertical Movement
If we move horizontally, the new vertex could be at:
- Horizontally right: \( (3 + 9, 5) = (12, 5) \)
- Horizontally left: \( (3 - 9, 5) = (-6, 5) \)
- Vertically up: \( (3, 5 + 9) = (3, 14) \)
- Vertically down: \( (3, 5 - 9) = (3, -4) \)
These give us four possible vertices:
- \( (12, 5) \)
- \( (-6, 5) \)
- \( (3, 14) \)
- \( (3, -4) \)
2. Diagonal Movement
If we move diagonally, the new vertex would be at a distance of \( 9\sqrt{2} \) because the diagonal of the square can be calculated by the relation \( d = s\sqrt{2} \), where \( d \) is the length of the diagonal and \( s \) is the side length.
However, it's simpler to recognize that for a square with a known vertex, the other vertices can be found by rotating the known vertex around the center of the square or simply moving from each vertex using the right triangle relationships, which leads us to derive new positions still confined by the right angles.
The diagonal movement gives us positions along the two possible 45-degree angles:
- By calculating new coordinates will yield the vertices accordingly, leading to positions like:
\[
(3 + 9\frac{\sqrt{2}}{2}, 5 + 9\frac{\sqrt{2}}{2}) = (3 + \frac{9\sqrt{2}}{2}, 5 + \frac{9\sqrt{2}}{2})
\] and others potentially reaching: - \( (3 - 9\frac{\sqrt{2}}{2}, 5 + 9\frac{\sqrt{2}}{2}) \)
- \( (3 - 9\frac{\sqrt{2}}{2}, 5 - 9\frac{\sqrt{2}}{2}) \)
- \( (3 + 9\frac{\sqrt{2}}{2}, 5 - 9\frac{\sqrt{2}}{2}) \)
These results, however, don't have integral coordinates straightforwardly. Thus returning to integer obvious calculations:
Conclusion
The possible coordinates of the other vertex (integral solutions) we derived mainly from easier straight movements lead mainly to those particular coordinates of:
- \( (12, 5) \)
- \( (-6, 5) \)
- \( (3, 14) \)
- \( (3, -4) \)
The derived possibilities should adhere primarily to straight distances: thus leading primarily the coordination solutions from horizontal/vertical edges as the major sources. So, final answer coordinates of another vertex could be:
- (12, 5), (-6, 5), (3, 14), or (3, -4).