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 m². But if these heights are 5 m and 7 m respectively, then their areas sum up to 123 m². Find the sum of the areas of the squares (in m²) drawn on the line segments.

Let the fixed widths of the two rectangles be $w_1$ and $w_2$.

Since width is fixed and only height varies, the area of a rectangle = width × height.

Setting up equations from the given conditions:

Condition	Rectangle 1 Area	Rectangle 2 Area	Total Area
Heights 7 m and 8 m	$7w_1$	$8w_2$	156 m²
Heights 5 m and 7 m	$5w_1$	$7w_2$	123 m²

This gives us two equations:

$7w_1 + 8w_2 = 156$ ... (1)

$5w_1 + 7w_2 = 123$ ... (2)

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Solving for the widths:

To eliminate $w_1$, multiply equation (1) by 5 and equation (2) by 7:

$35w_1 + 40w_2 = 780$

$35w_1 + 49w_2 = 861$

Subtracting the first from the second:

$9w_2 = 81$

$w_2 = 9$ m

Substituting $w_2 = 9$ in equation (2):

$5w_1 + 7(9) = 123$

$5w_1 + 63 = 123$

$5w_1 = 60$

$w_1 = 12$ m

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Finding the sum of areas of squares:

When squares are drawn on line segments of lengths 12 m and 9 m:

Area of square 1 = $(12)^2 = 144$ m²

Area of square 2 = $(9)^2 = 81$ m²

Sum = $144 + 81 = 225$ m²

Answer: Option 3