GATE 2023 EE GA Q.5: solution and answer

GATE 2023 · EEGA Q.5 · 1 mark · MCQ

How many triangles are present in the given figure?

GATE 2023 EE GA question 5: A parallelogram drawn in blue lines. — Figure from the GATE 2023 EE paper. It is part of the question.

(A)
12
(B)
16
(C)
20
(D)
24

Answer and solution

ANSWER

The figure contains 20 triangles, so the answer is option (C).

Official answer: CSource: Official answer key, GATE 2023 EE (IIT Kanpur)

Label the figure. Corners: $A$ bottom-left, $B$ bottom-right, $C$ top-right, $D$ top-left. Side midpoints: $Q$ on $A B$ , $R$ on $B C$ , $P$ on $C D$ , $L$ on $D A$ . Centre: $O$ , where the diagonals $A C$ and $B D$ cross. Diagonal $A C$ cuts $L Q$ at $E$ and $P R$ at $F$ . Diagonal $B D$ cuts $L P$ at $H$ and $QR$ at $G$ .
Sort the ten drawn lines by direction. The inner quadrilateral joins the midpoints of the sides, and the midpoint theorem says a segment joining the midpoints of two sides of a triangle is parallel to the third side. In triangles $A C D$ and $A B C$ this gives $L P ∥ A C ∥ QR$ . In triangles $A B D$ and $B C D$ it gives $L Q ∥ B D ∥ P R$ . So there are only four directions: horizontal ( $A B$ , $D C$ ), slant ( $A D$ , $B C$ ), the $A C$ direction ( $A C$ , $L P$ , $QR$ ) and the $B D$ direction ( $B D$ , $L Q$ , $P R$ ). Parallel lines never meet, so the three sides of every triangle must come from three different directions. Count the triangles in groups by size so that none is missed or counted twice.
Smallest triangles: a diagonal cuts each corner triangle in two. That gives 8: $A QE, A L E, D L H, D P H, C P F, C R F, B R G, B QG$ .
The four corner triangles that the inner parallelogram cuts off give 4 more: $A Q L, D L P, C P R, B QR$ .
Triangles with their apex at the centre $O$ give 4: $A O B, B O C, C O D, D O A$ .
Half-parallelograms, each cut off by one diagonal, give 4: $A B C, A C D$ (diagonal $A C$ ) and $A B D, B C D$ (diagonal $B D$ ).
No triangle lies inside the inner parallelogram $L P R Q$ . Every segment drawn inside it ( $L P$ , $QR$ and the piece $E F$ of $A C$ ; $L Q$ , $P R$ and the piece $H G$ of $B D$ ) runs in one of only two directions, and a triangle needs three. The diagonals just divide it into four small parallelograms, such as $L H O E$ .
Check that the list is complete by picking three of the four directions: horizontal, slant and $A C$ give $A B C, A C D, B QR, D L P$ (4). Horizontal, slant and $B D$ give $A B D, B C D, A Q L, C P R$ (4). Horizontal, $A C$ and $B D$ give $A O B, C O D, A QE, B QG, C P F, D P H$ (6). Slant, $A C$ and $B D$ give $A O D, B O C, A L E, D L H, C R F, B R G$ (6). These are the same 20 triangles.
Total: $8 + 4 + 4 + 4 = 20$ . The answer is (C).

KEY IDEA

Label every corner and crossing point, then sort the lines into parallel families: here there are four (horizontal, slant, and the two diagonal directions, with the inner parallelogram's sides parallel to the diagonals by the midpoint theorem). A triangle needs three sides from three different families. Count the triangles by size, then check the total by working through each choice of three families.

WATCH OUT FOR

Counting only the small triangles you can see at a glance and missing the large ones. If you leave out the four half-parallelograms ( $A B C$ , $A C D$ , $A B D$ , $B C D$ ) you get 16, which is option (B). If you also miss the four corner triangles ( $A Q L$ , $D L P$ , $C P R$ , $B QR$ ) that a diagonal splits in two, you get 12, which is option (A). Some students also look for triangles inside the inner parallelogram. There are none, because every line inside it runs in one of only two directions.

From the GATE 2023 EE paper. Every question of this paper · Download the full paper · All 23 Spatial questions, with notes

GATE 2023 EE GA Q.5: solution and answer

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