How many triangles are there in the following figure PQRS ?
[FIGURE: rectangle PQRS (P top-left, Q top-right, S bottom-left, R bottom-right) with both diagonals PR and QS drawn, plus a horizontal segm

Question

How many triangles are there in the following figure PQRS ?
[FIGURE: rectangle PQRS (P top-left, Q top-right, S bottom-left, R bottom-right) with both diagonals PR and QS drawn, plus a horizontal segment from the midpoint of side PS to the centre and a vertical segment from the centre down to side SR]

UnlockIAS · Accepted Answer

Correct — C, 12. Read the printed figure precisely first, because the whole question turns on how far the two extra lines run. Rectangle PQRS has both diagonals PR and QS, crossing at the centre O. A horizontal segment runs from M, the midpoint of the left side PS, rightward to O and stops there. A vertical segment runs from O downward to N, the midpoint of the bottom side SR, and stops there. Neither extra line crosses to the far side, so O, M and N are the only points added and the complete vertex list is P, Q, R, S, M, N, O — seven points, no other crossings anywhere. Now count in four disciplined groups instead of hunting by eye. (1) Each full diagonal cuts the rectangle into two halves: PR gives PQR and PRS, QS gives PQS and QRS — 4 triangles. (2) The two diagonals together carve four quadrant triangles around O, each standing on one full side: PQO on the top, QRO on the right, RSO on the bottom, PSO on the left — 4 more, running total 8. (3) The segment MO cuts the left quadrant PSO into PMO and SMO — 2 more, total 10. (4) The segment ON cuts the bottom quadrant RSO into SNO and RNO — 2 more, total 12. Nothing else qualifies: P–O–R and Q–O–S are straight lines, not triangles, and so are P–M–S and R–N–S. A full check of all 35 triples of the seven points confirms exactly these 12.

Why the other options are wrong — (a) Exactly what you get if you find one of the two subdivided quadrants and miss the other — most often you notice that MO splits the left quadrant into two, but overlook that ON splits the bottom quadrant the same way. Ten is the honest answer of a candidate who counted carefully but stopped one step early.  (b) The count you reach if you misread the two extra lines as complete mid-lines running right across the rectangle. Drawing the full horizontal and full vertical through O splits all four quadrants instead of two, adding four small triangles rather than two, and the total becomes exactly 16. It is the wrong figure, correctly counted.  (d) No reading of this figure produces 18 — even the fully mid-lined version tops out at 16. It sits above the correct answer to catch the candidate who assumes a busy-looking diagram must contain more triangles than a careful count found, and who therefore inflates rather than re-checks.

Concept: Figure-counting questions look like a test of eyesight but are really a test of method. Any such figure reduces to a set of marked points and the straight segments drawn between them, and a triangle exists for every set of three points that are (i) pairwise joined by drawn line, and (ii) not all on one straight line. Once the point set is fixed, the count is finite and provable rather than a matter of squinting. The reliable technique in the examination hall is to build up by size: first the triangles that use whole sides and whole diagonals, then the ones formed as those large triangles are cut by internal lines. Counting largest-first also gives a natural check, since every small triangle should be a piece of a larger one you have already listed.

State PCS 2024 Q92 — Science & Technology PYQ with Answer

State PCS 2024 Q92 — Science & Technology PYQ with Answer