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Step-by-Step Solution
Step 1: Represent the Unknown Coordinate of Vertex A
Since vertex A lies on the y-axis, let its coordinates be
$$(0,\,c).$$
Step 2: Identify the Lines Parallel to the Sides of the Rhombus
The rhombus has sides parallel to the given lines:
$$
x - y + 2 = 0 \quad \text{and} \quad 7x - y + 3 = 0.
$$
Written in slope-intercept form, these become:
$$
y = x + 2, \quad y = 7x + 3.
$$
Step 3: Relate Diagonals of the Rhombus to the Angle Bisectors
In a rhombus, the diagonals are along the angle bisectors of the lines that its sides are parallel to. The angle bisectors of
$$x - y + 2 = 0 \quad\text{and}\quad 7x - y + 3 = 0$$
can be found by the general formula for angle bisectors:
$$
\frac{x - y + 2}{\sqrt{1^2 + (-1)^2}}
\;=\; \pm \;
\frac{7x - y + 3}{\sqrt{7^2 + (-1)^2}}.
$$
Simplifying leads to lines that have slopes
$$m_1 = -\tfrac12 \quad\text{and}\quad m_2 = 2.$$
Step 4: Determine Which Diagonal Goes Through A and the Intersection Point P
The diagonals intersect at the point
$$(1,\,2).$$
One of these diagonals must pass through both
$$(0,\,c) \quad\text{and}\quad (1,\,2).$$
The slope of the line through these points is
$$
\frac{2 - c}{1 - 0} = 2 - c.
$$
Since a diagonal must match one of the slopes $- \tfrac12$ or $2,$ we set
$$
2 - c = 2 \quad \text{or} \quad 2 - c = -\tfrac12.
$$
Step 5: Solve for the Coordinate c
• If $\,2 - c = 2,\,$ then
$$c = 0,$$
which would place A at the origin, but the question states that A is different from the origin.
• If $\,2 - c = -\tfrac12,\,$ then
$$
-c = -\tfrac12 - 2 = -\tfrac52
\quad\Longrightarrow\quad
c = \tfrac52.
$$
Thus, the valid y-coordinate is
$$c = \tfrac{5}{2}.$$
Step 6: Conclude the Coordinates of Vertex A
Therefore, the vertex A, lying on the y-axis with y-coordinate $\,\tfrac{5}{2},\,$ is
$$
\bigl(0,\;\tfrac{5}{2}\bigr).
$$
Since the question’s correct answer format indicates the number $\tfrac{5}{2}$, we conclude:
$$
\boxed{\tfrac{5}{2}}
$$
