# In figure, $\mathrm{ABC}$ and $\mathrm{BDE}$ are two equilateral triangles such that $\mathrm{D}$ is the mid-point of $\mathrm{BC}$. If $\mathrm{AE}$ intersects $\mathrm{BC}$ at $\mathrm{F}$, show that(i) $\operatorname{ar}(\mathrm{BDE})=\frac{1}{4} \operatorname{ar}(\mathrm{ABC})$(ii) $\operatorname{ar}(\mathrm{BDE})=\frac{1}{2} \operatorname{ar}(\mathrm{BAE})$(iii) $\operatorname{ar}(\mathrm{ABC})=2$ ar $(\mathrm{BEC})$(iv) $\operatorname{ar}(\mathrm{BFE})=\operatorname{ar}(\mathrm{AFD})$(v) $\operatorname{ar}(\mathrm{BFE})=2 \operatorname{ar}(\mathrm{FED})$(vi) $\operatorname{ar}(\mathrm{FED})=\frac{1}{8} \operatorname{ar}(\mathrm{AFC})$[Hint: Join $\mathrm{EC}$ and $\mathrm{AD}$. Show that $\mathrm{BE} \| \mathrm{AC}$ and $\mathrm{DE} \| \mathrm{AB}$

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Given:

$\mathrm{ABC}$ and $\mathrm{BDE}$ are two equilateral triangles such that $\mathrm{D}$ is the mid-point of $\mathrm{BC}$.

$\mathrm{AE}$ intersects $\mathrm{BC}$

To do:

We have to show that

(i) $\operatorname{ar}(\mathrm{BDE})=\frac{1}{4} \operatorname{ar}(\mathrm{ABC})$
(ii) $\operatorname{ar}(\mathrm{BDE})=\frac{1}{2} \operatorname{ar}(\mathrm{BAE})$
(iii) $\operatorname{ar}(\mathrm{ABC})=2$ ar $(\mathrm{BEC})$
(iv) $\operatorname{ar}(\mathrm{BFE})=\operatorname{ar}(\mathrm{AFD})$
(v) $\operatorname{ar}(\mathrm{BFE})=2 \operatorname{ar}(\mathrm{FED})$
(vi) $\operatorname{ar}(\mathrm{FED})=\frac{1}{8} \operatorname{ar}(\mathrm{AFC})$

Solution:

Join $\mathrm{EC}$ and $\mathrm{AD}$

Let $G$ and $H$ be the mid-points of the sides $AB$ and $AC$ respectively.

Join $G$ and $H$.

$GH$ is parallel $BC$.

This implies, by mid point theorem,

$GH=\frac{1}{2}BC$

$GH=BD=DC$

Similarly,

$GD = AH = CH$

$DH = AG = BG$

This implies,

$\triangle ABC$ is divided into four equal equilateral triangles $\triangle BGD, \triangle AGH, \triangle DHC$ and $\triangle GHD$

Therefore,

$\triangle BGD=\frac{1}{4}\triangle ABC$

In $\triangle BDG$ and $\triangle BDE$

$BG=BE$

$BD=BD$

$DG=DE$

Therefore, by SSS congruency,

$\triangle BDG \cong \triangle BDE$

This implies,

$ar(\triangle BDG)=ar(\triangle BDE)$

$ar(\triangle BDE)=\frac{1}{4}ar(\triangle ABC)$

(ii)

$\triangle BDE$ and $\triangle AED$ have common base $DE$ and $DE \| AB$

This implies,

$ar(\triangle BDE) = ar(\triangle AED)$

$ar(\triangle BDE)−ar(\triangle FED) = ar(\triangle AED)−ar (\triangle FED)$

$ar(\triangle BEF) = ar(\triangle AFD)$.......…(i)

$ar(\triangle ABD) = ar(\triangle ABF)+ar(\triangle AFD)$

$ar(\triangle ABD) = ar(\triangle ABF)+ar(\triangle BEF)$         [From (i)]

$ar(\triangle ABD) = ar(\triangle ABE)$........…(ii)

$AD$ is the median of $\triangle ABC$.

This implies,

$ar(\triangle ABD) =\frac{1}{2}ar (\triangle ABC)$

$= 4\times[\frac{1}{2}ar (\triangle BDE)]$

$= 2 ar(ΔBDE)$.......…(iii)

From (ii) and (iii), we get,

$2 ar (\triangle BDE) = ar (\triangle ABE)$

$ar (\triangle BDE) =\frac{1}{2} ar (\triangle BAE)$

(iii) $\triangle ABE$ and $\triangle BEC$ have common base $BE$ and $BE \| AC$

This implies,

$ar(\triangle ABE) = ar(\triangle BEC)$

$ar(\triangle ABF) + ar(\triangle BEF) = ar(\triangle BEC)$

$ar(\triangle ABF) + ar(\triangle AFD) = ar(\triangle BEC)$             [From (i)]

$ar(\triangle ABD) = ar(\triangle BEC)$

$\frac{1}{2}ar(\triangle ABC) = ar(\triangle BEC)$

$ar(\triangle ABC) = 2 ar(\triangle BEC)$

(iv) $\triangle BDE$ and $\triangle AED$ lie on the same base $DE$ and between the parallels $DE$ and $AB$.

This implies,

$ar (\triangle BDE) = ar (\triangle AED)$

Subtracting $ar(\triangle FED)$ from both sides, we get,

$ar (\triangle BDE)−ar (\triangle FED) = ar (\triangle AED)−ar (\triangle FED)$

$ar (\triangle BFE) = ar(\triangle AFD)$

(v) Let $h$ be the height from $E$ to the side $BD$ in $\triangle BDE$ and $H$ the height from $A$ to the side $BC$ in $\triangle ABC$.

$ar (\triangle BDE) = \frac{1}{4}ar (\triangle ABC)$          (Proved)

$ar (\triangle BFE) = ar (\triangle AFD)$           (Proved)

$ar (\triangle BFE) = ar (\triangle AFD)$

$ar(\triangle BFE)= 2 ar (\triangle FED)$

(vi) $ar (\triangle AFC) = ar (\triangle AFD) + ar(\triangle ADC)$

$= 2 ar (\triangle FED) + \frac{1}{2}ar(\triangle ABC)$

$= 2 ar (\triangle FED) +\frac{1}{2}[4ar(\triangle BDE)]$

$= 2 ar (\triangle FED) +2 ar(\triangle BDE)$

$\triangle BDE$ and $\triangle AED$ are on the same base and between the same parallels= 2 ar (\triangle FED) +2 ar (\triangle AED)= 2 ar (\triangle FED) +2 [ar (\triangle AFD) +ar (\triangle FED)]= 2 ar (\triangle FED) +2 ar (\triangle AFD) +2 ar (\triangle FED)= 4 ar (\triangle FED) +4 ar (\triangle FED)ar (\triangle AFC) = 8 ar (\triangle FED)ar (\triangle FED) = \frac{1}{8}ar (\triangle AFC)\$

Hence proved.

Updated on 10-Oct-2022 13:46:30