If ABCD is a square, BD is parallel EF and EG = GF, which one is true?

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BE = FD

AG is angular bisector of BAD

AG if produced will pass through c

All of these

answer is D.

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Detailed Solution

Given that BD is parallel to EF,
which implies

\frac{CE}{EB}

\frac{CF}{FD}

\frac{x}{y}

Let us take CE = x and EB = y and CF = ax which implies FD = ay
which implies CB = x + y and CD = a (x + y)
As ABCD is a square CB should be equal to CD which implies a = 1
That says DF = BE = x, CE = CF = y.
So, the first option is correct.
Given that EG = GF
We know that ABCD is a square so ∠ABE = ∠ADF = 90°
And DE = BE and BA = AD
As both triangles, ABE and ADF are right-angled triangles as both the sides of two triangles are equal the remaining side should be equal.
which implies AE = EF.
Now as we know that all the sides of the triangles AEG and AGF are equal all the angles should be equal respectively which implies ∠EAG = ∠GAF
So AG is angular bisector of BAD
We know that ∠EAG = ∠GAF and ∠BAE = ∠FDA
which implies ∠EAG + ∠BAE = ∠GAF + ∠FDA => ∠BAE = ∠GAD
As ∠BAD is right angle ∠BAE = ∠GAD = 45°
Which says the line BG is angular bisector of the angle A.
In the square we know that the diagonal is itself the angular bisector of any angle, we can say that AG is diagonal.
Therefore, AG passes through C when extended.
So, all the options are correct.
Hence, Option (4) is the correct option.

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