Question

In the given figure, ABCD is a square, EF||BD and R is the mid point of EF. Prove that
i) BE=DF
ii) AR bisects angle BAD
iii) If AR is produced, it will pass through C​

Answer 1

Step-by-step explanation:

ABCD is a square and BD is a diagonal , so ∠CBD=45° .

EF//BD , so ∠CEF = ∠CBD = 45° .

And ∠ECF = 90° , so △ECF is an isosceles right triangle .

Therefore EC = FC . And BC = DC ,

so BE = BC - EC = DC - FC = DF . ---> 1. is proved .

Think △ABE and △ADF .

ABCD is a square so AB = AD and ∠ABD = ∠ADF = 90° .

And BE = DF , so they are congruent .

Therefore AE = AF and ∠BAE = ∠DAF .

Next , think △AER and △AFR .

AR = AR and AE = AF and ER = FR , so they are congruent .

Therefore ∠EAR =∠FAR .

So , ∠BAR = ∠BAE + ∠EAR = ∠DAF + ∠FAR = ∠DAR .

Therefore AR bisects ∠BAD . ---> 2. is proved .

ABCD is a square and AR bisects ∠BAD , so

AR is on a diagonal AC .

Therefore if AR is produced , it must pass through C . ---> 3. is proved .

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