Bit of a pain, but here goes:

1. Use the Pen tool and click on the point where you want your normal segment. See those two control handles coming from the line? The line they define is tangent to your curve.
   
   
2. Use the Pen tool and click exactly on one handle. Now you have one free point where one handle is.
   
   
3. Use the Direct Select tool (white arrow) and click again on the point on the curve.
   
   
4. Switch to the Pen tool and click exactly on the other control handle. Now you have two free points.
   
   
5. Selects the two point and go to Object > Path > Join.
   
   
6. With the segment still selected, switch to the Rotate tool and click on the point where the curve meets the segment.
   
   
7. Now hold Shift as you drag to rotate 90°.
   
   
8. Oh dear, there’s still that extra point you added to the curve. Let’s fix that. Select the new segment and Edit > Cut.
   
   
9. Undo repeatedly until you are back before step 1 (before the extra point was added).
   
   
10. Edit > Paste in Front.
    
    
11. Invent a new lexicon of curse words for 22 year old Adobe Illustrator, now at version 15 and still not the equal of programs half its age, selling for a fraction of its price.

RIdiculous, isn't it? Welcome to Illustrator.

For a single instance of constructing a perpendicular to a curve, you can streamline the above suggestions a bit by:

SmartGuides on. (SmartGuides is the only reasonably reliable snapping function in Illustrator.)

1. Select the curved path. Copy.

2. Add Anchor Point Tool (the 'plusPen'): Click the curved segment where you want the point of tangency. Do not deselect the path.

3. LineTool: Mousedown on one of the Direction Lines ('curve handles') of the added Anchor Point (SmartGuides snap to) and drag to the AnchorPoint (SmartGuides snap to).

4. DoubleClick the Rotate Tool and rotate 90° (or click the end of the Line [at the tangency anchorPoint] with the Rotate Tool to set the center of transformation there,  and then shiftDrag the opposite endPoint ).

5. If you don't want the added anchorPoint when you're done, delete the original curved path and then Paste In Front.

But for your fill pattern, you should build yourself a Pattern Swatch or a Brush, so you don't have to draw each perpendicular manually.

JET