Why is the small 8 by 8 square added while making the area model of a square with side x − 8?

Correct answer: It corrects the corner that was removed twice

Direct answer: A. Begin with a large square of area x². To represent side x − 8, remove a vertical strip of width 8 and a horizontal strip of width 8. The vertical strip has area 8x and the horizontal strip also has area 8x. However, the 8 by 8 corner belongs to both strips, so it has been removed twice. To correct this double removal, add that corner back once. Its area is 8 × 8 = 64. The identity is therefore (x − 8)² = x² − 8x − 8x + 64 = x² − 16x + 64. Option A explains the purpose of the added square. Option B is wrong because the 64 does not enlarge the original x²; it repairs the counting. Option C is wrong because the middle term comes from the two strips, giving −16x. Option D confuses area with boundary length; the small square does not represent perimeter. Memory cue: in (a − b)², subtract two ab regions but add b² back once.