Once you have the slope of the first line, remember that perpendicular lines have slopes whose product is $-1$. How can you use this to find the slope of line $n$?

Write the slope of line $n$ using its two points $(p,5)$ and $(9,-1)$ with the slope formula, and then set this equal to the perpendicular slope you found.

You will get a fraction involving $p$ set equal to a number. Clear the fraction by cross-multiplying and then isolate $p$.

In Desmos, type m1 = (-4-4)/(3-(-1)) to get the slope of the line through (-1,4) and (3,-4). Then type m2 = -1/m1 to get the slope of the line perpendicular to it.

In a new line, enter f(x) = (-1-5)/(9-x); this represents the slope of a line passing through (x,5) and (9,-1) as a function of x (this x plays the role of p).

On another line, type y = m2. Then add a graph of y = f(x). The x-coordinate of the intersection point of these two graphs is the value of p that makes the slopes equal. Read that x-value from Desmos.

Use the slope formula for the line through $(-1,4)$ and $(3,-4)$:

So the slope of the line through $(-1,4)$ and $(3,-4)$ is $-2$.

If two lines are perpendicular, the product of their slopes is $-1$.

Let the slope of line $n$ be $m_n$. Then

Solve for $m_n$:

So the slope of line $n$ must be $\tfrac{1}{2}$.

Line $n$ passes through $(p,5)$ and $(9,-1)$. Its slope is

This must equal the perpendicular slope you found in the previous step.

Set the two expressions for the slope of line $n$ equal:

Cross-multiply:

Now solve for $p$:

So the value of $p$ is $21$.