What would you do if you saw a lonely child at the playground?

I was at my kids’ school last Wednesday dropping off a Halloween non-treat (teachers requested a “healthy snack,” so we compiled a mobile taco bar). It was recess and, since I like to spy on my kids every chance I get, I secretly watched Milla play Four Square with her friends — which is what she tells me she does every evening when I ask her, “So, what did you do at recess today?”
 
Apparently, I make a terrible secret agent because, less than five seconds after I spotted her, Milla saw me and came running over. After we hugged (thank goodness she is not too cool to hug me in front of her friends), we started chatting, but I quickly got distracted by a girl behind her near the fence, pacing back and forth, alone. I tried to pay attention to what Milla was saying, but my eyes kept going back to the girl. She was very tall for her age, which I guessed was about 10, and had a sturdy build. Her long hair was a bit unkempt and she was wearing grey track pants and hoody. I noticed that her high tops were undone. She kept her head down as she walked about 10 paces along the fence, and then turned around and walked back the other way. She reminded me of a caged tiger at the zoo. My heart broke for her, and I was overwhelmed by that horrible feeling I still get when I think about my life in Grade 7.
 
“Who is that?” I asked Milla.

“I don’t know,” she said after watching the girl for a few seconds.
 
My head started spinning: Do I go over and talk to the girl? Would that be weird? What would I say? Do I ask Milla to go talk to her? How can we help her? Should I just walk away?
 
I just walked away. I crossed the parking lot, feeling like I was abandoning this poor little girl, who had no idea I existed or that I cared about her. Would I have made a difference if I had just reached out to her? Would my sympathy for her be welcomed, or would I be hurting her further by addressing her obvious unhappiness? Would she wonder who the crazy lady was if I’d tried to tell her I knew how she felt? Is it wrong to approach an unknown child and talk to them? I went about the rest of my day feeling like I had let the girl — and myself — down.
 
I am not making this part up: When I picked up Milla from school, she nonchalantly said, “Mommy, after you left, Hallie and I went up to that girl you saw. Her name is Monica. We asked her if she wanted to play Four Square with us. She said not today, but maybe tomorrow.”
 
I looked at my daughter and cried.
 
P.S. I thought long and hard before posting this, because I didn’t want people to think I was bragging about my daughter (who, I fully admit, made me extremely proud that day). It’s my goal that, by publishing this, you might share the story with your children, if they are old enough. Or, at the very least, I’d love to hear what you would have done if faced with a similar situation. Because I still don’t feel like I did the right thing, but thank goodness my daughter was there to save me.

What would you have done in a similar situation?