How many triangles are there in the figure given below?

Correct — C. Label the left edge T, M, N, B from the top.
P and Q end the upper and lower horizontals, R is the far-right tip, S is the foot of the short vertical under Q.
X is where the two middle slants cross.

Rule: list triangles by their three corners, and keep one only if every side is a drawn line.

Top: TMP, TNP = 2
Round X: MPX, MNX, NQX = 3
On the edge MN: MNP, MNQ = 2

Bottom: NQS, QRS, BNS = 3
Whole figure: BMR = 1
Total = 2 + 3 + 2 + 3 + 1 = 11 → option (c).

Why the others are wrong

• (a)
  13 — 13 is two more than the figure holds. Check each third side before counting: P, X and Q look closed, but no line joins P to Q.
• (b)
  12 — 12 adds one shape too many. N, Q and R look like a triangle, but nothing joins N straight to R; the slant from N runs down to S instead.
• (d)
  10 — 10 leaves one out. Check the largest last: BMR uses the whole edge from M down to B, the long slant from M to R and the bottom slant from B to R.

Triangle counting is an enumeration problem, not a spotting problem.

Label every corner and crossing first. Then list triangles by their three corners, keeping one only when all three sides are drawn lines and the three corners do not sit on one line.

Work in bands: the smallest cells first, then shapes made of two cells, then larger ones, ticking each so nothing is counted twice.

Key facts

• The figure has 9 straight lines: 2 verticals, 2 horizontals and 5 slants.
• Three lines bound a triangle only when their three meeting points are distinct and lie on drawn segments.
• The 11 triangles are TMP, TNP, MPX, MNX, NQX, MNP, MNQ, NQS, QRS, BNS and BMR.

Common traps

• Counting only the smallest cells and stopping there.
• Counting a three-sided-looking region whose third side is not drawn.
• Counting a large triangle twice, once whole and once as its two halves.

No directly related past PYQ was found.