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8  triangles
There is 1 large triangle, 2 small triangles, 3 long triangles inside 1 medium triangle, and another medium triangle.  I think I got all of the triangles.

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Whole Triangle
Inside top
3 Top &  1 Inside
Inside 2
Inside 3
All the top
Top 3
2 of the top
2 of the top
Top 3
2  Top & 2 Inside
2 Top & 2 Inside
Inside and Top
Bottom
3 on Bottom
2 Bottom
Middle row
Middle and Bottom
Top 3
Top and Middle

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I counted 9 triangles. There is one large triangle (the outline). Then, I saw 3 medium triangles placed inside the large one. Next, I found one small one on the tip of the medium middle triangle. I found 3 skinny triangles on the left side inside of the medium triangle. Finally, I counted 1 more on the top of the other side of the medium triangle. I counted 9 triangles, but I'm unsure if I found them all. :)

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Okay this is going to seem very complex, but I see 19. The big one that makes up the entire picture, 3 long ones on the right, then when you look at the middle I see the small one, under the small one I believe that makes up 3 more when broken down then at the bottom in the middle I see 3 more. Now to the right side of the big triangle I see at the top that long shape that it makes up 4 triangles, underneath that makes up 3 and then the small one under that. So in all that should be 19 triangles. I hope that makes sense because it’s hard to explain.

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23 triangles
* whole triangle
Starting on the left of the triangle
*top
*middle
*bottom
*top/middle combo
*middle/bottom combo
*top/middle/bottom combo
long triangles
*top
*middle
*bottom
*top /middle combo
*middle/bottom combo
triangle in middle
*whole triangle
*top
*top/middle
*front top/middle top combo
*front middle/middle middle combo
*front bottom/bottom middle combo
*front top/middle  and middle/top middle
*top middle bottom and middle middle bottom
*small triangle bottom right
*small triangle and quad middle
*small triangle bottom.quad middle and quad top

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I see 18 triangles.
*1 large triangle

*3 long triangles going from top to bottom

*3 triangles from left to right

*3 triangles in front section

*1 small triangle in top

*1 small triangle in bottom right

*1 middle triangle- small triangle and quad.

*1 triangle on left side made of 2 small triangles- top

*1 triangle on left side made of 2 small triangles- bottom

*1 triangle on right side from small triangle and quad.

*1 triangle- 2 long triangles, 1 small triangle, and quad.

*1 triangle- 2 long triangles, 2 quad. shapes

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I counted 15 triangles. The large one, Three long triangles in the middle plus the 3 that the whole is divided into. 5 little triangles all on the inside. Plus 3 more that take up two smaller ones in the middle.

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I got 22 triangles
3 small on the left then combine them to make 2 more using 2 triangles.
Then extend them to make 3 medium length then 3 more long ones. This totals 15 triangles.
Next I took the middle section and counted 3 more triangles to total 18. Then I counted the right side and made 3 more triangles and one more around the entire perimeter. The total is 22.

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While I was able to count the triangles, I found it very difficult to describe how I counted them. I drew a color coded picture to help.
1=big triangle; 4= yellow, blue, green shaded triangle, plus yellow and blue together; 9= purple lined triangles (3 from yellow section only, 3 from yellow and blue section only, 3 from yellow, blue and green section); 6= combining purple lined triangles from yellow, blue, and green sections; 2= blue section; 2= green section. All together that equals 24 triangles.

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I counted 24! I labeled each vertex so I could track what 'letters' I had already used. This is what I got:
1. ABI
2. AIH
3. AHG
4. ABH
5. AIG
6. ABG
7. ABJ
8. AJK
9. AKF
10. ABK
11. AJF
12. ABF
13. ACD
14. ADE
15. AEF
16. ACE
17. ADF
18. ACF
19. BIJ
20. BHK
21. BGF
22. FEK
23. FDJ
24. FBC

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I counted 9 triangles.

1 big
3 skinny
2 small
3 medium

clearly by reading the various comments...I am probably wrong  but I honestly don't understand some of the vocabulary in some of the responses.  I'm glad I only teach 2nd grade...:)