In ∆ PQR, M, N, and S are the mid-points of sides PQ, PR and QR, respectively. If the area of the ∆ PQR is 46 cm 2 , what is the area (in cm 2 ) of the quadrilateral MNRQ?
- (a)39
- (b)34.5
- (c)23
- (d)27.6
Answer
Why
Correct — B. M and N are the midpoints of PQ and PR, so MN is the midsegment: MN is parallel to QR and MN = QR ⁄ 2.
That makes ∆PMN similar to ∆PQR with sides in 1 : 2, so their areas are in 1 : 4.
Area of ∆PMN = 46 ÷ 4 = 11.5 cm²
MNRQ is the rest of the triangle:
46 − 11.5 = 34.5 cm² → option (b).
S, the midpoint of QR, never enters the calculation — it lies on side RQ, but it is not a vertex of MNRQ.
Why the others are wrong
- (a)39 — Thirty-nine matches neither piece. The midsegment cuts ∆PQR into exactly two parts, a small triangle of 11.5 cm² and the trapezium MNRQ of 34.5 cm², and there is nothing else to measure.
- (c)23 — Twenty-three is half of 46. A midsegment does not halve the area — it cuts off a quarter, because the areas of similar triangles go as the square of the side ratio, and (1⁄2)² = 1⁄4.
- (d)27.6 — 27.6 is exactly three-fifths of 46. The midsegment splits this triangle in the ratio 1 : 3, giving 11.5 and 34.5. No division of this figure lands in fifths.
Concept
Join the midpoints of two sides of a triangle and you get a midsegment: parallel to the third side and half its length.
The small triangle it cuts off is similar to the whole, with k = 1⁄2, so its area is k² = 1⁄4 of the original — and the quadrilateral left behind is the other 3⁄4.
Drawing all three midsegments makes this concrete: they cut any triangle into four congruent triangles, each a quarter of the whole. MNRQ is three of those four.
The question defines S, the midpoint of QR, and then asks about MNRQ, which is bounded by MN, NR, RQ and QM. S is on RQ but is not a vertex.
An unused given is a standard SSC decoration. Confirm which shape the question names before you use every letter offered.
Key facts
- Midpoint theorem: the segment joining the midpoints of two sides is parallel to the third side and half its length.
- The three midsegments cut any triangle into four congruent triangles, each one quarter of the whole.
- MNRQ is a trapezium, since MN is parallel to QR and equals half of it.
- Areas of similar triangles are in the square of the ratio of their corresponding sides.
Study next
Common traps
- Halving the area because the segment joins two midpoints
- Treating MNRQ as a parallelogram, when MN and QR are unequal
- Forcing the unused midpoint S into the working
SSC gives one area figure, a midpoint set-up, and asks for the leftover quadrilateral rather than the triangle you naturally compute. The same square law is asked plainly in this shift at Quant Q.4, where areas of 25 : 144 give sides of 5 : 12.
Related PYQs
No directly related past PYQ was found.