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ABCD is a square. E, F, and G are midpoints of side AB, BC, CD respectively. Then is it true that the triangles AEF and DGF are congruent?

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True

False

answer is A.

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Detailed Solution

Given, ABCD is a square. E, F, and G are midpoints of side AB, BC, CD respectively.
Question Image

△ EBF

and

△ FCG

,
BF = CF (Since F is the midpoint of BC)
EB = GC (Since ABCD is a square and E, G are the midpoints of AB, CD respectively)
∠EBF = ∠FCG = 90°
So,

△ EBF ⩭ △ FCG

[By SAS congruence rule].
Thus, EF = GF since the two triangles are similar.
Now, in

△ ABF

and

△ DCF

,
BF = CF (Since F is the midpoint of BC)
AB = DC (Since ABCD is a square)
∠ABE = ∠DCF = 90°
So,

△ ABF ⩭

△ DCF

. [By SAS congruence rule].
Thus, AF = DF since the two triangles are similar.
Now, in

△ AEF

and

△ DGF

,
EF = GF (From above calculations)
AF = DF (From above calculations)
AE = DG (From above calculations)
So,

△ AEF ⩭ △ DGF

.
So, the given statement is true and the correct option is 1.

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