How many triangles are there in the given figure?

Correct — D. SSC's tentative key marks 8, and this page is written to that key — read the note before you settle on a total of your own.

Rule: label every corner and every crossing, then count in families. A crossing is a vertex like any drawn corner.

A tilted quadrilateral overlaps a pentagon. Four lines are added across them:

the quadrilateral's top corner to the pentagon's left corner,
that left corner across to the quadrilateral's right corner,
the pentagon's apex down to the same right corner,
and a long chord from the quadrilateral's left corner to the pentagon's bottom-right.

Upper family. The quadrilateral's right corner and the pentagon's left corner are both four-line junctions, and the wedges between them close the top half.

Lower family. The long chord cuts the pentagon's left edge and the quadrilateral's lower-right edge, which cut each other too — three crossings, with a small triangle between them.

Why the others are wrong

• (a)
  11 — Eleven is above any reading of this drawing. The quadrilateral's outline and the pentagon's outline close no triangle between them, so every triangle here needs one of the added chords, and the chords do not create that many.
• (b)
  7 — Seven drops at least one closure. Both four-line junctions — the quadrilateral's right corner and the pentagon's left corner — carry thin wedges that a quick scan skips.
• (c)
  9 — Nine is not the total SSC's key marks. A strict segment-by-segment enumeration does reach it — that is what the note on this card records — but the answer the exam scored is 8.

Figure counting is the reasoning item where a system beats speed. Label every corner and every crossing, then work through the lines instead of scanning the picture for shapes.

Two rules keep the count honest. A triangle needs three drawn segments that close on each other, and a crossing point counts as a vertex exactly like a drawn corner.

Then count in families: triangles made by the outlines alone, triangles that use one added chord, and triangles that sit wholly inside a cluster of crossings. Families are what stop you counting a shape twice.

Key facts

• A crossing of two drawn lines counts as a vertex, exactly like a corner of the figure.
• Every triangle in this drawing uses at least one of the added chords. The two outlines close none between them.
• SSC's tentative key for this question marks 8.

Common traps

• Counting the triangles you can see at a glance and stopping there.
• Missing a small triangle formed where two chords cross a third line.
• Counting a four-sided region as a triangle because a line passes close to one corner.

No directly related past PYQ was found.