How many rectangles are present in the given figure? The figure shows a square with both diagonals drawn, creating an X pattern inside the square, and also with the mid-lines (horizontal and vertical) drawn, dividing the square into smaller regions. A. 8 B. 9 C. 10 D. 12
GATE 2023 · General Aptitude · Patterns In Two Dimensions · medium
Answer: There are 10 rectangles in the given figure — option C.
- Label the regions: The figure has 8 triangular sub-regions (4 corner triangles and 4 inner triangles meeting at the centre). Label them T1 (top), T2 (right), T3 (bottom), T4 (left) as the outer triangles, and t1, t2, t3, t4 as the inner triangles adjacent to the centre.
- Count rectangles using horizontal/vertical lines only: Lines: 3 horizontal (top edge, mid, bottom edge) and 3 vertical (left edge, mid, right edge). Rectangles from choosing 2 horizontal and 2 vertical lines = C(3,2) x C(3,2) = 3 x 3 = 9.
- Check for additional diagonal-enabled rectangles: The diagonals create additional intersection points. Checking all combinations: the two half-diagonal strips (NW-SE diagonal split and NE-SW diagonal split) each form a rectangle when taken as strips across the full square. These are counted within the 9 from step 2. One additional rectangle: the centre diamond formed by connecting the four mid-edge points — this is a square rotated 45 degrees, which is indeed a rectangle (all angles 90 degrees). Count: 9 + 1 = 10.
- Final count: Total rectangles = 10. This matches option C.