A675-meter bike trail passes through forest and meadows. there are six legs of the trail that cut through meadows: two legs measure meters, three legs measure 52.25 meters, and one leg measures 32 meters. the rest of the trail passes through forest. how many meters of the trail pass through the forest?

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step-by-step explanation:

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4