Question
The base $BC$ of $\triangle\text{ABC}$ is divided at $D$ such that $\text{BD}=\frac{1}{2}\text{DC}.$ Prove that $\text{ar}(\triangle\text{ABD})=\frac{1}{3}\times\text{ar}(\triangle\text{ABC}).$
✓
Answer
Given: $D$ is a point on $BC$ of $\triangle\text{ABC},$ such that $\text{BD}=\frac{1}{2}\text{DC}$
To prove: $\text{ar}(\triangle\text{ABD})=\frac{1}{3}\times\text{ar}(\triangle\text{ABC}).$
Construction: Draw $\text{AL}\perp\text{BC}.$
Proof: In $\triangle\text{ABC},$
we have: $BC = BD + DC \Rightarrow BD + 2BD = 3 \times BD$
Now, we have: $\text{ar}(\triangle\text{ABD})=\frac{1}{2}\times\text{BD}\times\text{AL}$
$\text{ar}(\triangle\text{ABC})=\frac{1}{2}\times\text{BC}\times\text{AL}$
$\Rightarrow\ \text{ar}(\triangle\text{ABC})=\frac{1}{2}\times3\text{BD}\times\text{AL}$
$\Rightarrow\ 3\times\Big(\frac{1}{2}\times\text{BD}\times\text{AL}\Big)$
$\Rightarrow\ \text{ar}(\triangle\text{ABC})=3\times\text{ar}(\triangle\text{ABD})$
$\therefore\ \text{ar}(\triangle\text{ABD})=\frac{1}{3}\text{ar}(\triangle\text{ABC})$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The following table gives the distribution of $IQ's$ (intelligence quotients) of $60$ pupils of class $V$ in a school:
Draw a frequence polygon for the above data.
→|
IQ's:
|
$125.5$ to $13.25$
|
$118$ to $125.5$
|
$111.5$ to $118.5$
|
$104.5$ to $111.5$
|
$97.5$ to $104.5$
|
$90.5$ to $97.5$
|
$83.5$ to $90.5$
|
$76.5$ to $83.5$
|
$69.5$ to $76$
|
$62.5$ to $69.5$
|
|
No. of pupils:
|
$1$
|
$3$
|
$4$
|
$6$
|
$10$
|
$12$
|
$15$
|
$5$
|
$3$
|
$1$
|
Two chords $AB$ and $CD$ of a circle intersect each other at $P$ outside the circle. If $AB = 6\ cm, BP = 2\ cm$ and $PD = 25\ cm$, find $CD$. 
→If $\text{x}=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$ and $\text{y}=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ then find the value of $x^2 + y^2$.
→Find rational numbers $a$ and $b$ such that: $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=\text{a}+\text{b}\sqrt{3}$
→In a cyclic quadrilateral $ABCD$, if $(\angle\text{B}-\angle\text{D})=60^\circ,$ show that the smaller of the two is $60^\circ$
→Read the following bar graph and answer the following questions:
$i.$ What information is given by the bar graph?
$ii.$ In which year the export is minimum?
$iii.$ In which year the import is maximum?
$iv.$ In which year the difference of the values of export and import is maximum?
→→$i.$ What information is given by the bar graph?
$ii.$ In which year the export is minimum?
$iii.$ In which year the import is maximum?
$iv.$ In which year the difference of the values of export and import is maximum?
The image of an object placed at a point A before a plane mirror $LM$ is seen at the point B by an observer at $D$ as shown in figure. Prove that the image is as far behind the mirror as the object is in front of the mirror.
→In the following, using the remainder theorem, find the remainder when $f(x)$ is divided by $g(x)$ and verify the by actual division: $f(x) = 9x^3 - 3x^2 + x - 5$, $\text{g(x)}=\text{x}-\frac{2}{3}$
→Evaluate:$ (28)^3 + (-15)^3 + (-13)^3$
→