Question

The area of a rectangle is 45 cm². Two squares are constructed such that two adjacent sides of the rectangle are each also the side of one of the squares. The combined area of the two squares is 106 cm². Find the lengths of the sides of the squares.

Answer 1

5 cm, 9 cm

Explanation:

If x and y are the lengths of sides of the rectangle, then the problem statement tells you ...

• xy = 45
• x² +y² = 106

Substituting y = 45/x into the second equation, you get

... x² + (45/x)² = 106

Multiplying by x² gives ...

... x⁴ -106x² +2025 = 0 . . . . . after subtracting 106x²

... (x² -53)² -784 = 0 . . . . . . . . complete the square

... x² = 53 ±√784 = 53 ±28 . . . solve for x²

... x = √81 or √25 = 9 or 5 . . . . solve for x (which must be positive)

y is the other of these two values, so ...

The lengths of the sides of the squares are 5 cm and 9 cm.

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Comment on the graph

The graph is symmetrical about the origin, so there are also two solutions at (x, y) = (-5, -9) and (-9, -5). Since x and y represent distances, these are extraneous solutions with respect to the original question.