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Question

E and F are the midpoints of the sides AB and AC respectively of ΔABC; G and H are the midpoints of sides AE and AF respectively of the ΔAEF. If GH=1.8 cm, find BC.


A

0.9 cm

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B

3.6 cm

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C

7.2 cm

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D

5.4 cm

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Solution

The correct option is B

7.2 cm


Given:GH=1.8 cm,To find BC.

Also:AE=EB and AF=FC (E & F are midpoints of AB & AC)

AG=GE and AH=HF(G & H are midpoints of AE & AF)

GH || EF BC (line joining the midpoints of two sides of a triangle is parallel to the third side)

& (if a || b and b || c, then a || c)

Given:In Δ AGH and AEF:

1. HAG FAE (common angle)

2. AGH=AEF (corresponding angles : GH || EF with AE as a transversal)

3. GHA=EFA (corresponding angles: GH || EF with AF as a transversal) So Δ AGH Δ AEF(by A-A-A similarity criterion)

By similarity criterion :

AG : AE=GH : EF

1:2=1:8:EF(G is the midpoint of AE)

EF=(2×1.8)=3.6 cm

Similarly, In Δ AEF and Δ ABC:

1. FAE=CAB (common angle)

2. AEF=ABC (corresponding angles: EFBC with AB as a transversal)

3. EFA=BCA (corresponding angles: EFBC with AC as a transversal)

So Δ AGH Δ AEF (by A-A-A similarity criterion)

By similarity criterion :

AE:AB=EF:BC

1:2=3.6:BC (E is the midpoint of AB)

BC=(2×3.6)

=7.2 cm


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