How many triangles are there in the given figure?

Correct — A. Name the points first, then count by base.

The picture is a large triangle standing on a horizontal base. Call the base corners A (left) and B (right), and the apex T. A vertical line runs from T down to M, the middle of the base.

A point P sits on that vertical, below T. Lines AP and BP are drawn, so a smaller roof sits inside the big one. Six segments in all: AB, AT, BT, TM, AP, BP.

Rule: count every triangle by the drawn line that forms its base.

On the full base AB: ATB and APB — 2
On the half-bases AM and MB: AMT, BMT, AMP, BMP — 4
On the upper part of the vertical, TP: ATP and BTP — 2

2 + 4 + 2 = 8 → option (a).

Why the others are wrong

• (b)
  9 — 9 needs a ninth shape with all three sides drawn, and there is none. The tempting extra is a triangle spanning AP and BT, but those two lines never meet inside the figure.
• (c)
  11 — 11 counts shapes that are not closed by drawn segments. Test each candidate against the six lines actually present — AB, AT, BT, TM, AP and BP — and only eight pass.
• (d)
  10 — 10 usually comes from counting the two inner slants twice, once with the near slant and once with the far one. AP meets AT only at A, so it forms no triangle with BT.

Figure counting is only reliable when it is systematic. The safest method is to count by base: list the drawn segments, then for each one count the triangles that rest on it, working from the longest base to the shortest.

Every triangle must have all three of its sides present as drawn segments. A shape whose third side is merely implied by the eye is not a triangle in this question.

Because each triangle is counted once, under the base you assigned it, the method also removes the commonest source of error, which is counting the same shape twice under two names.

Key facts

• Count by base: take each drawn segment in turn and count the triangles resting on it.
• A triangle counts only if all three sides are drawn segments.
• This figure has six drawn segments — AB, AT, BT, the vertical TM, and the inner slants AP and BP.
• The eight are ATB, APB, AMT, BMT, AMP, BMP, ATP and BTP.

Common traps

• Missing ATP and BTP, the two that use only the upper piece of the vertical.
• Counting one triangle twice because it can be described from two different corners.

No directly related past PYQ was found.