Answer:
30(√3 +√2 +1) ≈ 124.3879
Explanation:
Given the figure with triangle ABC and segments HI, FG, DE parallel to BC dividing the triangle into four equal areas, you want to know the sum of the lengths of DE, FG, and HI.
Scale factor
The scale factor for linear dimensions of similar figures is the square root of the scale factor for area.
∆AHI ~ ∆ABC has 1/4 the area, so HI = BC·√(1/4)
∆AFG ~ ∆ABC has 1/2 the area, so FG = BC·√(1/2) = BC·√(2/4)
∆ADE ~ ∆ABC has 3/4 the area, so DE = BC·√(3/4)
Sum
The sum DE +FG +HI is then ...
DE +FG +HI = BC·(√(3/4) +√(2/4) +√(1/4)) = 60(√3 +√2 +1)/2
DE +FG +HI = 30(√3 +√2 +1) ≈ 124.3879
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