    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: EF∥BC with AB as a transversal) 3. ∠EFA=∠BCA (corresponding angles: EF∥BC 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  Suggest Corrections  0      Similar questions  Explore more