Geodesist

Triangle shaped field (triangle ABC) has side AB = 129 m. path XY is parallel to the side AB which divided triangle ABC into two parts with same area.

What will be the length of the path XY? Help please geodesist ...

Correct result:

|XY| =  91.2 m

Solution:

$c = c_2 = 129\ m \ \\ 2 S_1 = S_2 \ \\ \ \\ S_1 = \dfrac{ c_1 h_1 }{ 2 } \ \\ S_2 = \dfrac{ c_2 (h_1 + h_2) }{ 2 } \ \\ \ \\ \dfrac{c_1}{h_1}= \dfrac{ c_2 }{ h_1 + h_2} \ \\ h_1 + h_2 = \dfrac{ c_2 h_1 }{c_1} \ \\ \ \\ 2 \dfrac{ c_1 h_1 }{ 2 } = \dfrac{ c_2 (h_1 + h_2) }{ 2 } \ \\ 2 c_1 h_1 = c_2 (h_1 + h_2) \ \\ 2 c_1 h_1 = c_2 \dfrac{ c_2 h_1 }{c_1} \ \\ \ \\ 2 c_1^2 h_1 = c_2^2 h_1 \ \\ 2 c_1^2 = c_2^2 \ \\ c_1^2 = c_2^2 / 2 \ \\ c_1 = c_2 / \sqrt{2} = 129 / \sqrt{2} = 91.2 \ \text{m}$

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