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CBSE Class 10 Answered

ABCD is a parallelogram with diagonals AC & BD.Prove that AC squared ➕ BC squared=ABsq. ➕ BC sq. ➕ CD sq. ➕ DA sq.
Asked by agraharianiruddha | 26 Jan, 2019, 06:16: PM
answered-by-expert Expert Answer
Figure shows a parallelogram ABCD.
Let us extend AB to point E and draw perpendicular line CE. Let begin mathsize 12px style angle end styleBEC = 90º. Let us also draw line BF perpendicular to CD.
in right ΔACE , we have AC2 = AE2+EC2  = (AB+BE)2 + EC2 = AB2 + BE2 +2×AB×BE + EC2 = AB2 + BC2 +2×AB×BE .......................(1)
in right ΔBFD , we have BD2 = BF2+DF2  = BF2 + (CD - CF)2  = BF2 + CD2 + CF2 - 2×CD×CF = BC2+CD2 -2×CD×CF.........................(2)
 
If we add eqn.(1) and (2), terms  '2×AB×BE' and '2×CD×CF' will be nullified, because AB = CD abd BE = CF.
 
also, since we have BC = AD, if we add eqn.(1) and eqn.(2), we get AC2 + BD2 = AB2 + BC2 + CD2 + AD2
Answered by Thiyagarajan K | 26 Jan, 2019, 11:28: PM

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