a city has a park shaped as a right angled traingle . the length of the longest side of this park is 80 m. the mayor of the city wants toconstruct 3 paths from the corner point opp to the longest side such that these paths divide the longest side into 4 equal segments . determine the sum of squares of the lengths of three paths​

Dear student

If right triangle triad 3,4,5 is considered and the proportionate values to suite length of diagonal to 80, the triad becomes: 48,64,80The other two sides are 48 and 64.

BC=80, AB=64, AC=48From trignomteric ratios,B=sin-14880 =sin-135orcos-16480=cos-145From right triangle BIF,BF=BI cosB=20×45=16AF=AB-BF=64-16=48FI=BI sin B=20×35=12From right triangle BEH,BE=BH cosB=40×45=32AE=AB-BE=64-32=32EH=BHsinB=40×35=24From right triangle BDGBD=BGcosB=60×45=48AD=AB-BD=64-48=16DG=BGsinB=60×35=36Applying pythagoras to the right triangle AIFAI2=AF2+FI2=482+122Applying pythagoras to the right triangle AHEAH2=AE2+EH2=322+242Applying pythagoras to the right triangle AGDAG2=AD2+DG2=162+362The sum of the squares of the lengths of three paths=AI2+AH2+AG2=482+122+322+242+162+362=5600 m

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