1. Given lines forming two adjacent sides

The two adjacent sides of parallelogram $ABCD$ lie on $2x-3y=-23$ and $5x+4y=23.$

Their intersection gives the vertex $A$.

So solve: $2x-3y=-23 \quad (1)$ $5x+4y=23 \quad (2)$

From (1): $2x=-23+3y$ $x=\frac{-23+3y}{2}.$

Substitute into (2): $5\left(\frac{-23+3y}{2}\right)+4y=23$ $\frac{-115+15y}{2}+4y=23$ $-115+15y+8y=46$ $23y=161$ $y=7.$

Then $2x-3(7)=-23$ $2x-21=-23$ $2x=-2$ $x=-1.$

Hence, $A=(-1,7).$

2. Use the fact that diagonals of a parallelogram bisect each other

One diagonal is $AC$ whose equation is $3x+7y=23.$

Since $A=(-1,7)$, $3(-1)+7(7)=-3+49=46\neq 23.$

So this is impossible if the diagonal is really $AC$, because diagonal $AC$ must pass through $A$.

Thus there is evidently a typo in the statement: the given diagonal must be the other diagonal, i.e. $BD$.

Then the question asks for distance of $A$ from the other diagonal, which would then be $AC$; but only one diagonal equation is given. The natural interpretation consistent with geometry and the stored answer is:

• the given line $3x+7y=23$ is one diagonal,
• we need distance of $A$ from that diagonal.

So let us compute distance of $A$ from the line $3x+7y-23=0.$

3. Distance of point $A(-1,7)$ from the line $3x+7y-23=0$

Using point-to-line distance formula, $d=\frac{|3(-1)+7(7)-23|}{\sqrt{3^2+7^2}}.$

Compute numerator: $3(-1)+49-23=-3+49-23=23.$

Denominator: $\sqrt{9+49}=\sqrt{58}.$

Thus $d=\frac{23}{\sqrt{58}}.$

So, $d^2=\frac{529}{58}.$

Therefore, $50d^2=50\cdot \frac{529}{58}=\frac{25\cdot 529}{29}.$

Since $529=23^2$ and $58=2\cdot 29$, $50d^2=\frac{26450}{58}=\frac{13225}{29}.$

Now, $29\cdot 456=13224,$ so $50d^2=\frac{13225}{29}\neq 529.$

This shows again that the printed statement is inconsistent as written.

4. Check what value matches the stored answer

If instead the intended quantity were $58d^2,$ then $58d^2=529.$

This matches the natural distance computation from $A$ to the given line.

So the most likely issue is a typo in the question: either the coefficient $50$ is wrong, or the named diagonal is wrong.

5. Conclusion

From the geometry and the given line, the actual computed value is $d=\frac{23}{\sqrt{58}}, \qquad 50d^2=\frac{13225}{29}.$

This is not equal to the stored answer $529$.

Hence I do not agree with the stored answer under the question as written.