Download App

Areas Of Parallelograms And Triangles — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsAreas Of Parallelograms And Triangles5 Marks
Question
$\text{ABCD}$ is a parallelogram whose diagonals intersect at $O$ .If $P$ is any point on $BO,$ prove that:
$i. \text{ar}(\triangle\text{ADO})=\text{ar}(\triangle\text{CDO})$
$ii. \text{ar}(\triangle\text{ABP})=\text{ar}(\triangle\text{CBP})$
✓

Answer

Given that $\text{ABCD}$ is the parallelogram To Prove:
$i. \text{ar}(\triangle\text{ADO})=\text{ar}(\triangle\text{CDO}).$
$ii. \text{ar}(\triangle\text{ABP})=\text{ar}(\triangle\text{CBP}).$
Proof: we know that diagonals of parallelogram bisect each other
$\therefore AO = OC$ and $BO = OD$
$i.$ In $\triangle\text{DAC},$ since $DO$ is a median.
Then $\text{ar}(\triangle\text{ADO})=\text{ar}(\triangle\text{CDO}).$
$ii.$ In $\triangle\text{BAC},$ since $BO$ is a median.
Then $\text{ar}(\triangle\text{BAO})=\text{ar}(\triangle\text{BCO})\ ....(1)$
In $\triangle\text{PAC},$ since $PO$ is a median.
Then $\text{ar}(\triangle\text{PAO})=\text{ar}(\triangle\text{PCO})\ .....(2)$
Subtract equation $2$ from $1.$
$\Rightarrow\text{ar}(\triangle\text{BAO})−\text{ar}(\triangle\text{PAO})\\ \ =\text{ar}(\triangle\text{BCO})−\text{ar}(\triangle\text{PCO})$
$\Rightarrow\text{ar}(\triangle\text{ABP})=2 \text{ar}(\triangle\text{CBP}).$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Areas Of Parallelograms And Triangles→Areas Of Parallelograms And Triangles — all question types→Maths STD 9 question papers→Gujarat Board STD 9 question papers→Gujarat Board question paper generator→

Similar questions

Bisectors of interior $\angle\text{B}$ and exterior $\angle\text{ACD}$ of a $\Delta\text{ABC}$ intersect at the point $T.$ Prove that,
$\angle\text{BTC}=\frac{1}{2}\angle\text{BAC}.$
→
The value of $\frac{(0.013)^3+(0.007)^3}{(0.013)^2-0.013\times0.007+(0.007)^2},$ is:
→
In the given figure, $AB \| CD \| EF , \angle D B G=x, \angle E D H=y, \angle A E B=z, \angle E A B=90^{\circ}$ and $\angle B E F=65^{\circ}$. Find the values of $x , y$ and $z$ .Image
→
In a circle of radius $5\ cm, AB$ and $CD$ are two parallel chords of lengths $8\ cm$ and $6\ cm$ respectively. Calculate the distance between the chords if they are:
$i.$ On the same side of the centre.
$ii.$ On the opposite sides of the centre.
→
The diagonals of a quadrilateral $\text{ABCD}$ are perpendicular. Show that the quadrilateral, formed by joining the mid$-$points of its sides, is a rectangle.
→
A sphere and a right circular cylinder of the same radius have equal volumes. By what percentage does the diameter of the cylinder exceed its height$?$
→
In the given figure, $O$ is the centre of the circle, $BD = OD$ and $\text{CD}\perp\text{AB}.$ Find $\angle\text{CAB}.$
→
In the given figure, $AB \| CD$. Prove that $p + q - r =180$.
Image
→
$1500$ families with $2$ children were selected randomly, and the following data were recorded:
No of girls in a family $0$ $1$ $2$
No of girls $211$ $814$ $475$
If a family is chosen at random, compute the probability that it has:
$1.$ No girl.
$2. 1$ girl.
$3. 2$ girls.
$4.$ At most one girl.
$5.$ More girls than boys.
→
In the given figure, $\triangle\text{ABC}$ is equilateral. Find
$i. \angle\text{BDC}$
$ii. \angle\text{BEC}$
→