Question

E and F are the midpoints of the sides AB and AC respectively of $\Delta ABC$; G and H are the midpoints of sides AE and AF respectively of the $\Delta AEF.$ If GH=1.8 cm, find BC.

• A. 0.9 cm
• B. 3.6 cm
• C. 7.2 cm
• D. 5.4 cm

Answer 1

The correct option is C

7.2 cm

$Given:GH=1.8cm,TofindBC.$

$Also:AE=EBandAF=FC\left(E\&FaremidpointsofAB\&AC\right)$

$AG=GEandAH=HF\left(G\&HaremidpointsofAE\&AF\right)$

$GH||EFBC\text{(line joining the midpoints of two sides of a triangle is parallel to the third side)}$

$\&(ifa||bandb||c,thena||c)$

$Given:In\Delta AGHandAEF:$

$1.\angle HAG\angle FAE\left(commonangle\right)$

$2.\angle AGH=\angle AEF\text{(corresponding angles : GH}||EFwithAEasatransversal)$

$3.\angle GHA=\angle EFA(correspondingangles:GH||EFwithAFasatransversal)$ $\text{So}\Delta AGH\Delta AEF\text{(by A-A-A similarity criterion)}$

$\text{By similarity criterion :}$

$\text{AG : AE}=\text{GH : EF}$

$\Rightarrow 1:2=1:8:EF\text{(G is the midpoint of AE)}$

$\Rightarrow EF=(2\times 1.8)=3.6cm$

$\text{Similarly,}In\Delta AEFand\Delta ABC:$

$1.\angle FAE=\angle CAB\text{(common angle)}$

$2.\angle AEF=\angle ABC\text{(corresponding angles}\text{: EF}\parallel \text{BC with AB as a transversal)}$

$3.\angle EFA=\angle BCA\text{(corresponding angles}\text{: EF}\parallel \text{BC with AC as a transversal)}$

$\text{So}\Delta AGH\Delta AEF\text{(by A-A-A similarity criterion)}$

$\text{By similarity criterion :}$

$AE:AB=EF:BC$

$\Rightarrow 1:2=3.6:BC\text{(E is the midpoint of AB)}$

$\Rightarrow BC=(2\times 3.6)$

$=7.2cm$