A circle touches the side's EF of triangle DEF at P and touches sides DE and DF at Q and R respectively. show that DQ=1/2(perimeter of triangle DEF
Answers
Given : A circle touches the side's EF of triangle DEF at P and touches Extended sides DE and DF at Q and R respectively.
To Find : show that DQ=1/2(perimeter of triangle DEF)
Solution:
perimeter of Δ DEF = DE + DF + EF
EF = EP + PF
=> perimeter of Δ DEF = DE + DF + EP + PF
EP = EQ Equal tangents
FP = FR Equal tangents
=> perimeter of Δ DEF = DE + DF + EQ + FR
=> perimeter of Δ DEF = DE+ EQ + DF + FR
=> perimeter of Δ DEF = DQ + DR
DQ = DR Equal tangents
=> perimeter of Δ DEF = DQ + DQ
=> perimeter of Δ DEF = 2DQ
=> (1/2) perimeter of Δ DEF = DQ
=> DQ = (1/2) perimeter of Δ DEF
QED
hence proved
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