Question: A hydrologist designs a triangular reservoir with vertices at $ (0, 0, 0) $, $ (4, 0, 0) $, and $ (0, 4, 0) $. Find the coordinates of the fourth vertex to complete a parallelogram, ensuring all coordinates are integers. - Treasure Valley Movers

A hydrologist designs a triangular reservoir with vertices at $ (0, 0, 0) $, $ (4, 0, 0) $, and $ (0, 4, 0) $. Find the coordinates of the fourth vertex to complete a parallelogram, ensuring all coordinates are integers.

In an era where spatial design and predictive modeling shape infrastructure planning, a growing interest in geometric logic behind water storage systems surfaces—particularly in the precise layout of triangular reservoirs. This question reflects a broader trend: professionals seeking mathematical clarity to optimize land and resource use. Users across the U.S. are exploring how spatial geometry informs resilient, efficient design—driven by sustainability goals and urban development needs.

Why This Question is Gaining Traction in the US
Modern water infrastructure demands smarter planning, and spatial reasoning plays a key role. Professionals and planners increasingly use coordinate geometry to visualize reservoir expansions, especially when transitioning from triangular forms to parallelograms for enhanced capacity and flow dynamics. This question taps into a niche but expanding community of hydrological planners studying coordinate-based transformation techniques—valued for accuracy in site layout and compliance with environmental modeling standards.

How to Complete the Parallelogram: A Clear Explanation
A parallelogram in coordinate geometry is defined by opposite sides being equal and parallel. Given three vertices $ A(0, 0, 0) $, $ B(4, 0, 0) $, and $ C(0, 4, 0) $, the fourth vertex $ D $ can be found by vector addition. The vector $ \vec{AB} = (4, 0) $ and $ \vec{AC} = (0, 4) $. Adding $ \vec{AB} $ to point $ C $, or $ \vec{AC} $ to $ B $, gives:
$ D = B + \vec{AC} = (4, 0) + (0, 4) = (4, 4, 0) $
or
$ D = C + \vec{AB} = (0, 4) + (4, 0) = (4, 4, 0) $

This confirms that the fourth vertex is $ (4, 4, 0) $, an integer-coordinate point forming a symmetrical, scalable parallelogram.

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