Two rectangles are drawn on line segments of fixed widths. When the heights are 7 m and 8 m respectively, their areas sum up to 156 $$m^{2}$$. But if these heights are 5 m and 7 m respectively, then their areas sum up to 123 $$m^{2}$$. Find the sum of the areas of the squares (in $$m^{2}$$) drawn on the line segments.

Let u be the fixed width of the first rectangle and v be the width of the second rectangle.

Given,

1. $$7u + 8v = 156$$

2. $$5u + 7v = 123$$

Multiplying equation 1. by 5 and equation 2. by 7, we get

3. $$35u + 40v = 780$$
4. $$35u + 49v = 861$$

On solving this system, we get
$$v = 9$$
$$u = 12$$

Sum of areas of squares drawn on the line segments
$$= u^{2 }+ v^2$$
$$=12^{2 }+ 9^2$$
$$= 225$$

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