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CBSE Class 10 Answered

M is any point outside the square ABCD and MA is greater than MB. Prove that MA^2-MB^2=MD^2-MC^2.
Asked by rushabhjain.av | 30 Jan, 2019, 09:33: PM
answered-by-expert Expert Answer
Figure shows a poin M outside square. let us draw lines CF extending side DC and BE extending side AB.
Join MF and ME so that MF perpendicular to CF and ME perpendicular to BE.
 
from ΔAME, MA2 = AE2 + ME2 = (AB+BE)2 - ME2 = AB2 + BE2 + 2×AB×BE + ME2 ..................(1)
 
from ΔBME,   MB2 = ME2 + BE2 .........................................(2)
from (1) and (2),    MA2 - MB2 = AB2 + 2×AB×BE .............................(3)
 
from ΔMFD, MD2 = MF2+ FD2 = MF2 + (DC+CF)2 = MF2 +DC2 +CF2 +2×DC×CF ........................(4)
from ΔMCF, MC2 = MF2 +FC2 .......................(5)
from (4) and (5), MD2 - MC2 = DC2 + 2×DC×CF .......................(6)
 
since AB = DC and BE = CF, from (4) and (6),  MA2 - MB2 = MD2 - MC2
Answered by Thiyagarajan K | 31 Jan, 2019, 11:46: AM

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