In the standard \((\mathrm{x},\mathrm{y})\) coordinate plane, the graph of |3x| + |7y| = 21 is a parallelogram with 2 vertices...

Let's visualize this problem to make it crystal clear.

Phase 1: Understanding the Shape

The equation \(|3\mathrm{x}| + |7\mathrm{y}| = 21\) creates a special quadrilateral. Let's find its vertices by determining where it crosses the axes.

Finding x-axis intersections (where y = 0):

• \(|3\mathrm{x}| + |7(0)| = 21\)
• \(|3\mathrm{x}| = 21\)
• \(|\mathrm{x}| = 7\)
• So \(\mathrm{x} = 7\) or \(\mathrm{x} = -7\)
• Vertices on x-axis: \((7, 0)\) and \((-7, 0)\)

Finding y-axis intersections (where x = 0):

• \(|3(0)| + |7\mathrm{y}| = 21\)
• \(|7\mathrm{y}| = 21\)
• \(|\mathrm{y}| = 3\)
• So \(\mathrm{y} = 3\) or \(\mathrm{y} = -3\)
• Vertices on y-axis: \((0, 3)\) and \((0, -3)\)

Phase 2: Visualizing the Parallelogram

Let's draw this parallelogram:

       (0,3)
         |
         |
(-7,0)---+---(7,0)  
         |
         |
       (0,-3)

The four vertices form a rhombus (special parallelogram) centered at the origin.

Phase 3: Finding the Diagonal Lengths

Horizontal diagonal H:

• Connects \((-7, 0)\) to \((7, 0)\)
• \(\mathrm{H} = 7 - (-7) = 14\) coordinate units

Vertical diagonal V:

• Connects \((0, -3)\) to \((0, 3)\)
• \(\mathrm{V} = 3 - (-3) = 6\) coordinate units

Phase 4: Solution

From our analysis:

• \(\mathrm{H} = 14\) (horizontal diagonal length)
• \(\mathrm{V} = 6\) (vertical diagonal length)

Answer:

• For H: Select 14
• For V: Select 6