Scaling linear dimensions by factor k increases area by k²: 2² = 4 and 3² = 9. Yes, in each case exactly 4 or 9 congruent copies tile and perfectly fit into the scaled figure without gaps.
Three problems about fitting congruent shapes together: (i) Rectangle ABCD has sides a, b and rectangle PQRS has sides 2a, 2b. Show that PQRS has 4 times the area of ABCD. Does this mean that 4 copies of rectangle ABCD will fit into rectangle PQRS? Check and see! (ii) ΔABC has sides a, b, c and ΔPQR has sides 2a, 2b, 2c. Show that ΔPQR has 4 times the area of ΔABC. Does this mean that 4 copies of ΔABC will fit into ΔPQR? Check and see! (iii) ΔABC has sides a, b, c and ΔPQR has sides 3a, 3b, 3c. Show that ΔPQR has 9 times the area of ΔABC. Does this mean that 9 copies of ΔABC will fit into ΔPQR? Check and see!
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(i) Area(ABCD) = ab. Area(PQRS) = (2a) x (2b) = 4ab = 4 x Area(ABCD).
By dividing each side of PQRS into two equal parts, it divides into a 2-by-2 grid of 4 identical copies of ABCD that fit perfectly.
(ii) By Heron’s formula or similarity, scaling sides by 2 scales area by 2² = 4.
Joining the midpoints of the sides of ΔPQR creates the medial triangle and divides ΔPQR into 4 congruent copies of ΔABC, which fit seamlessly.
(iii) Scaling sides by 3 scales area by 3² = 9.
Trisecting each side of ΔPQR and drawing grid lines parallel to the sides subdivides ΔPQR into 1 + 3 + 5 = 9 identical copies of ΔABC that fit completely without overlap.
For more NCERT Solutions of Class 9 Ganita Manjari Chapter 6 Measuring Space: Perimeter and Area Question Answer (2026-27)
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