How many triangles are there in the given figure?

Correct — D.

Rule: a triangle can only appear where a slanting line cuts a cell. The plain rectangles contribute none, so visit only the six places a slant is drawn.

Top-left square: the inscribed diamond cuts off the four corner triangles.

Top band: two long slants meet in a V on the vertical beneath them, giving the two halves and the whole — 3.

Right edge: the arrowhead's upper and lower slants each close against the right-hand edge — 2.

The two single-diagonal cells in the middle: one diagonal halves each — 2 + 2.

Bottom left: the lower slant halves a closed rectangle (2); the upper slant closes only downwards, because no horizontal runs this far left above it (1); and the two slants together bound one larger triangle (1) — 4.

That is seventeen you can be sure of.

The inscribed diamond is drawn a hair clear of the lines that cross it — at its lower-left side and again at its right vertex — and each near-miss clips off one more sliver of a triangle. The key marks 18 → option (d).

Why the others are wrong

• (a)
  12 — 12 is short by six. The bottom-left corner alone accounts for four triangles and the top band for three, which is seven before the top-left square is looked at.
• (b)
  17 — 17 is the sweep with every clipped corner treated as coincident and ignored — four in the top-left square and thirteen elsewhere. The key counts at least one of those clips, so 17 lands one short of what is marked.
• (c)
  15 — 15 is short by three. Each of the two middle cells gives two triangles rather than one, and the bottom-left corner gives four, because its two slants together bound a large triangle on top of the smaller ones.

Counting figures reward a fixed route rather than a stare. Two rules cut the work down.

First, a triangle needs a slanting side. Every triangle here has one, so the grid of plain rectangles can be ignored and only the slants need visiting.

Second, count in layers: the smallest triangles each slant makes, then any built by joining two of them. Two places in this figure produce such a composite.

Those are the V where two long slants meet on a vertical, and the bottom-left corner, where two slants leave the same point on the left edge and a vertical closes them off at their far ends.

Key facts

• Every triangle in this figure has at least one slanting side, so the rectangles alone contribute none.
• The inscribed diamond cuts four corner triangles from the top-left square.
• Two long slants meeting on a vertical give three triangles: two halves and the whole.
• The bottom-left corner gives four triangles — two from the lower slant, one from the upper, and one from the pair together.

Common traps

• Counting only the smallest triangles and missing the ones made from two
• Counting a shape twice when a line passes through more than one cell
• Spending more than a minute on a counting item whose options sit one apart

No directly related past PYQ was found.