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Saturday, December 29, 2018

Area: Trapezium


Parallel sides of a trapezium are equal to a and b . Find the length of the line segment which is parallel to them and divides the area of the trapezium into two equal parts.
( i.e Find x in terms of a and b .)


Solution

Draw AD and BC produced meet at G .

\displaystyle \text{Let}\ \alpha (\Delta GDC)=P\ ,\ \alpha (DEFG)=Q\ \ ,\ \alpha (ABFE)=RLet α(ΔGDC)=P , α(DEFG)=Q  , α(ABFE)=R 

\displaystyle \Delta GDC\sim \Delta GEF\sim \Delta GAB\ \ (\because DC\parallel EF\parallel AB) 

\displaystyle \frac{P}{{P+Q}}=\frac{{{{b}^{2}}}}{{{{x}^{2}}}} 

\displaystyle \frac{P}{Q}=\frac{{{{b}^{2}}}}{{{{x}^{2}}-{{b}^{2}}}}\ ........(1)

\displaystyle \frac{P}{{P+Q+R}}=\frac{{{{b}^{2}}}}{{{{a}^{2}}}} 

\displaystyle \frac{P}{{Q+R}}=\frac{{{{b}^{2}}}}{{{{a}^{2}}-{{b}^{2}}}}

\displaystyle \frac{P}{{2Q}}=\frac{{{{b}^{2}}}}{{{{a}^{2}}-{{b}^{2}}}}\ \ (\because \ Q=R) 

\displaystyle \frac{P}{Q}=\frac{{2{{b}^{2}}}}{{{{a}^{2}}-{{b}^{2}}}}\ ........(2)

\displaystyle \text{From}\ (1)\ \text{and}\ (2)\ ,\ \frac{{{{b}^{2}}}}{{{{x}^{2}}-{{b}^{2}}}}\ =\frac{{2{{b}^{2}}}}{{{{a}^{2}}-{{b}^{2}}}} 

\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2{{x}^{2}}-2{{b}^{2}}={{a}^{2}}-{{b}^{2}} 

\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2{{x}^{2}}={{a}^{2}}+{{b}^{2}} 

\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \therefore \ \ \ x=\sqrt{{\frac{{{{a}^{2}}+{{b}^{2}}}}{2}}}

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