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Question

In ABC ,Given that DE//BC , D is the midpoint of AB and E is a midpoint of AC. The ratio AE : EC is ____.


A

1:2

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B

2:1

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C

1:1

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D

None of the above

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Solution

The correct option is C

1:1


DE is parallel to BC

So, In triangles ABC , ADE

DAE = ECF {Alternate angles}

ADE = EFC {Alternate angles}

∠BAC = ∠DAE
By A.A.A similarity ABC≡ADE
⇒ADDB=AEEC ( Basic Proportionality Theorem )Since, D is midpoint of AB .AD=DB⇒ADDB=11=AEEC⇒AEEC=1
ALTERNATE SOLUTION

D is the midpoint of AB. A line segment DE is drawn which meets AC in E and is parallel to the opposite side BC.

BCFD is a parallelogram DF || BC and CF || BD

CF = BD {opposite sides of parallelogram are equal}

CF = DA {Since BD = DA given}

In ADE and CFE

AD = CF

DAE = ECF {Alternate angles}

ADE = EFC {Alternate angles}

ADE ≡ CFE

AE = EC {Corresponding parts of congruent triangles are equal}

AE: EC = 1: 1



In triangles ADE


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