Two right triangles with common hypotenuse form quadrilateral that lying in the square with all 4 vertices on square's sides, as shown in the picture. Find the exact value of the length of the side of the square.
Answer 192.
Let the angle between side 7 and the vertical line be x
=> angle between side 24 and horizontal line = x
and angle between side 20 and vertical line
= 180° - x - arctan(24/7) - arctan(3/4)
= arctan(117/44) - x = y - x ... [Taking arctan(117/44) = y]
arctan (117/24) = arcsin(117/125) = arccos(44/125)
=> siny = 117/125 and cosy = 44/125
=> 7sinx + 24cosx = 7cosx + 20cos( y - x)
=> 7sinx + 17cosx = 20cosx cosy + 20sinx siny
=> (20siny - 7) sinx = (17 - 20cosy) cosx
=> tanx = (17 - 20cosy) / (20siny - 7)
=> tanx = [17 - 20 * (44/125)] / [20 * (117/125) - 7] = 249/293
=> sinx = 249/[5√(5914)] and cosx = 293/[5√(5914)]
=> Exact length of the side of the square
= 7sinx + 24cosx
= 7 * 249 / [5√(5914) + 24 * 293 / [5√(5914)]
= 1755 / √(5914)
≈ 22.82.
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