### Find areas in composite triangles and parallelogram by considering pairs of similar triangles and triangles of same height

Question Sample Titled 'Find areas in composite triangles and parallelogram by considering pairs of similar triangles and triangles of same height'

In the figure, ${A}{B}{C}{D}$ is a parallelogram. ${E}$ is a point lying on ${C}{D}$ such that ${D}{E}:{E}{C}={4}:{5}$. ${A}{D}$ produced and ${B}{E}$ produced meet at ${F}$ while ${A}{E}$ produced and ${B}{C}$ produced meet at ${G}$. If the area of $\triangle{D}{E}{F}$ is ${448}$ $\text{cm}^{{2}}$, then the area of $\triangle{C}{E}{G}$ is

${A}$${B}$${C}$${D}$${F}$${G}$${E}$
A
${875}$ $\text{cm}^{{2}}$ .
B
${560}$ $\text{cm}^{{2}}$ .
C
${700}$ $\text{cm}^{{2}}$ .
D
${1008}$ $\text{cm}^{{2}}$ .

 Consider $\triangle{D}{E}{F}$ and $\triangle{C}{E}{B}$ , $\angle{D}{E}{F}$ $=\angle{C}{E}{B}$ vert. opp. ∠s $\angle{E}{D}{F}$ $=\angle{E}{C}{B}$ alt. ∠s ${D}{F}$$//$${B}{C}$ $\angle{D}{F}{E}$ $=\angle{C}{B}{E}$ alt. ∠s ${D}{F}$$//$${B}{C}$ ∴  $\triangle{D}{E}{F}$ ~ $\triangle{C}{E}{B}$ AAA ∴  Area of $\triangle{C}{B}{E}={448}\times{\left(\dfrac{{5}}{{4}}\right)}^{{2}}={700}$ $\text{cm}^{{2}}$  Consider $\triangle{C}{B}{E}$ and $\triangle{D}{E}{A}$ . Since ${A}{B}{C}{D}$ is a parallelogram, so height of $\triangle{C}{B}{E}$ $=$ height of $\triangle{D}{E}{A}$ Area of $\triangle{D}{E}{A}$ $={\left(\dfrac{{4}}{{5}}\right)}\times{(}$area of $\triangle{C}{B}{E}{)}$ $={\left(\dfrac{{4}}{{5}}\right)}\times{700}$ $={560}$ $\text{cm}^{{2}}$  Consider $\triangle{D}{E}{A}$ and $\triangle{C}{E}{G}$ , $\angle{D}{E}{A}$ $=\angle{C}{E}{G}$ vert. opp. ∠s $\angle{E}{D}{A}$ $=\angle{E}{C}{G}$ alt. ∠s ${A}{F}$$//$${B}{G}$ $\angle{D}{A}{E}$ $=\angle{C}{G}{E}$ alt. ∠s ${A}{F}$$//$${B}{G}$ ∴  $\triangle{D}{E}{A}$ ~ $\triangle{C}{E}{G}$ AAA ∴   Area of $\triangle{C}{E}{G}={560}\times{\left(\dfrac{{5}}{{4}}\right)}^{{2}}={875}$ $\text{cm}^{{2}}$

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