CBSE Class 9 Answered
prove that the line segment joining the middel point of the sides of a triangle divides it into four congruent triangles
Asked by sariatiw123 | 02 Dec, 2018, 03:25: PM
Expert Answer
Figure shows a ΔABC and D, E and F are the repective middle points of sides AB, BC and AC .
Line DF is drawn through middle points of side AB and side AC. Hence DF || BC . Hence 
ADF = ABC and AFD = ACB.
ADF = ABC and AFD = ACB. Let ADF = ABC = β (marked in figure) and AFD = ACB = γ (marked in figure)
ADF = ABC = β (marked in figure) and AFD = ACB = γ (marked in figure)Similarly EF || AB, we can consider CFE = CAB and CEF = CBA .
CFE = CAB and CEF = CBA . Let CFE = CAB = α and CEF = CBA = β (marked in figure)
CFE = CAB = α and CEF = CBA = β (marked in figure)Similarly DE || AC, we can consider BDE = BAC and BED = BCA .
BDE = BAC and BED = BCA . Let BDE = BAC = α and BED = BCA= γ (marked in figure)
BDE = BAC = α and BED = BCA= γ (marked in figure)BED + DEF + FEC = 180°. Hence DEF = 180° - ( DEB + FEC ) = α [ 
α, β and γ are angles of ΔABC, hence α + β + γ = 180° ]Similarly we can prove EFD = β and FDE = γ .
EFD = β and FDE = γ .Hence all the triangles are having similar angles α, β and γ.
Also all the triangles have respective equal sides. For example BE = EC = DF [ E is middle point of BC and DF = (1/2)BC = BE ]
E is middle point of BC and DF = (1/2)BC = BE ]Hence as seen from figure, ΔADF, ΔBDE, ΔEFC and ΔDEF are congruent .
Answered by Thiyagarajan K | 02 Dec, 2018, 11:34: PM
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