Four nested squares as shown in the figure. (A square inside another square is obtained by joining its midpoints.) If the largest square is 2 inches on one side, can you find a relationship between the areas of the squares? Also find a relationship between the length of their sides.
Answers
Solution :-
Let us assume that,
- Largest square outside is denoted as 1 .
- Second square inside is denoted as 2 .
- Square with red sides is denoted as 3 .
- Square with blue area is denoted as 4.
so,
→ Side of square 1 = 2 inches .
→ Area of square 1 = (2)² = 4 inches² .
now, since square 2 is formed by joining the mid points .
→ side of square 2 = √(1² + 1²) = √2 inches . { since all angles of a square are 90° and side of square 2 is hypotenuse of right angled ∆ so formed with base and perpendicular at mid points of square 1.}
→ Area of square 2 = (√2)² = 2 inches² .
similarly,
→ side of square 3 = √[(√2/2)² + (√2/2²] = [(2/4) + (2/4)] = 1 unit .
→ Area of square 3 = (1)² = 1 inches² .
and,
→ side of square 4 = √[(1/2)² + (1/2)²] = √[(1/4 + 1/4)] = √(1/2) = (1/√2) inches .
→ Area of square 4 = (1/√2)² = (1/2) inches² .
therefore, we can conclude that,
→ Relationship between the areas of the squares = Area of squares are in GP with common ratio as (1/2) . { 4, 2, 1, 1/2 }
→ Relationship between the sides of the squares = Sides of squares are in GP with common ratio as √2 . { 2, √2, 1, 1/√2 }
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