Mathematics

Parallelogram Properties and Area

93 Questions

Parallelogram properties and area questions test the ability to calculate dimensions using base and height ratios. Problems also cover diagonal properties and internal angles. These geometry concepts regularly appear in quantitative aptitude sections of various exams.

Area calculationBase and height ratiosDiagonal propertiesInterior anglesGeometric theorems

Parallelogram Properties and Area Questions

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

if $\hat { i } +2\hat { j } +3\hat { k } $ and $ 3\hat { i } -2\hat { j } +\hat { k } $ are the adjacent sides of a parallelogram, then its area will be 

  1. $8\sqrt { 3 } $
  2. $5\sqrt { 3 } $
  3. $16\sqrt { 3 } $
  4. $6\sqrt { 3 } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We have,

$ \overrightarrow{a}=\widehat{i}+2\widehat{j}+3\widehat{k} $

$ \overrightarrow{b}=3\widehat{i}-2\widehat{j}+\widehat{k} $


We know that,

Area of parallelogram $=\left| \begin{matrix} i & j & k \\ 1 & 2 & 3 \\ 3 & -2 & 1 \end{matrix} \right| $

$ =\left( -6-2 \right)\widehat{i}-\widehat{j}\left( 9-1 \right)+\widehat{k}\left( 6+2 \right) $

$ =-8\widehat{i}-8\widehat{j}+8\widehat{k} $


$ Now, $

$ \left| \overrightarrow{a}\times \overrightarrow{b} \right|=\sqrt{{{\left( -8 \right)}^{2}}+{{\left( -8 \right)}^{2}}+{{8}^{2}}} $

$ =\sqrt{64+64+64} $

$ =\sqrt{3\times 64} $

$ =8\sqrt{3}\,\,sq.\,unit $


Hence, this is the answer.

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

ABCD is a parallelogram with sides $AB = 12\ cm$, $BC = 10\ cm$ and diagonal $AC = 16\ cm$. Find the approximate area of the parallelogram.

  1. $119.8cm^2$
  2. $103.7cm^2$
  3. $15.7cm^2$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Area of triangle with sides $12\ cm, 10\ cm, 16\ cm$:
$s=\dfrac{12+10+16}{2}=19$

Area, A = $\sqrt{s(s-a)(s-b)(s-c)}$

$A=\sqrt{19(19-12)(19-10)(19-16)}$

$A=59.9$ sq. cm

Therefore,
Area of parallelogram $=2A = 2\times 59.9 = 119.8$ sq. cm
Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

Two opposite angles of a parallelogram are $(3x-2)^{\circ}$ and $(50-x)^{\circ}$. Find the measure of each angle of the parallelogram.

  1. $40^{\circ},140^{\circ},40^{\circ},140^{\circ}$
  2. $37^{\circ},143^{\circ},37^{\circ},143^{\circ}$
  3. $35^{\circ},145^{\circ},35^{\circ},145^{\circ}$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since opposite angles of a parallelogram are equal. Therefore,
$3x-2=50-x\Rightarrow x=13$


$(3x−2)^{\circ}=3(13)-2=37^{\circ}$

The measures of the adjacent angles of a parallelogram add up to be $180$ degrees, or they are supplementary.
Another angle $=180-37=143^{\circ}$

The measure of each angle of the parallelogram.
$37^{\circ},143^{\circ},37^{\circ},143^{\circ}$

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

Find the measure of all the angles of a parallelogram, if one angle is $24^{\circ}$ less than twice the smallest angle.

  1. $68^{\circ},112^{\circ},68^{\circ},112^{\circ}$
  2. $48^{\circ},72^{\circ},48^{\circ},72^{\circ}$

  3. Insufficient data

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the smallest angle be of $x^{\circ}$. 


Then, the other angle is of $(2x-24)^{\circ}$. 

Since adjacent angles of a parallelogram are supplementary.

$\therefore x^{\circ}+(2x-24)^{\circ}=180^{\circ}\Rightarrow x=68^{\circ}$


$(2x-24)^{\circ}=2(68)-24=112^{\circ}$


The measure of all the angles of a parallelogram are
$68^{\circ},112^{\circ},68^{\circ},112^{\circ}$

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

In a parallelogram ABCD, if $ \angle B = 135^{\circ},$ determine the measures of its other angles.

  1. $ \angle A = 45^{\circ}, \angle C = 45^{\circ},\angle D = 135^{\circ} $
  2. $ \angle A = 135^{\circ}, \angle C = 45^{\circ},\angle D = 45^{\circ} $
  3. $ \angle A = 40^{\circ}, \angle C = 50^{\circ},\angle D = 135^{\circ} $

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We have, $\angle B = 135^{\circ} $ 


Since ABCD is a parallelogram.

$ \therefore \angle A = \angle C, \angle B = \angle D $ and $ \angle A + \angle B = 180^{\circ} $

$ \Rightarrow \angle A = \angle C = 45^{\circ} $ and $ \angle B = \angle D = 135^{\circ} $

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram
$ABCD$ is a parallelogram of area $162\ sq. cm$. $P$ is a point on $AB$ such that $AP : PB = 1 : 2$. 
Calculate the ratio $PA : DC$.
  1. $1 : 3$
  2. $3:1$
  3. $3:2$
  4. $2:3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given,

$ABCD$ is a parallelogram, Area of parallelogram $ABCD=162cm^2$

$P$ is such a point on $AB$ such that $AP:PB=1:2$

Now, $\frac{AP}{PB}=\frac{1}{2}=k$[Let]

Thus, $AP=k$ and $PB=2k$

And, $AB=AP+PB$

   $=>AB=k+2k$

$=>AB=3k$

$\therefore \frac{AP}{AB}=\frac{1}{3}$

Now, $AB=CD$ [Opposite sides of a parallelogram are equal]

$\therefore \frac{AP}{CD}=\frac{1}{3}$


Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram
$ABCD$ is a parallelogram of area $162\: cm^2$. $P$ is a point on $AB$ such that $AP : PB = 1 : 2$. Calculate the area of $\Delta APD$
  1. $20\; cm^2$
  2. $27\ cm^2$
  3. $24\ cm^2$
  4. $25\ cm^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If $DB$ is the diagonal,
Area of parallelogram $\displaystyle \frac{1}{2}ABCD =$ area $ADB = $ area $BDC$
Area ADB$=\displaystyle \frac{1}{2} (162)=81$
$P$ is mid point of $AB$ in the ratio $1:2$ $ \left (\dfrac{1}{3}+\dfrac{2}{3}\right)$ 

Area APD $=\displaystyle \frac{1}{3}\times $ Area of $ADB=\displaystyle \dfrac{1}{3}(81)=27 $ sq. cm

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

The area of a parallelogram is $y$ $cm^{2}$ and its height is $h\ cm$. The base of another parallelogram is $x\ cm$ more than the base of the first parallelogram and its area is twice the area of the first. Find, in terms of $y, h$ and $x$, the expression for the height of the second parallelogram.

  1. $\displaystyle \frac{hy}{yh-x}$
  2. $\displaystyle \frac{y}{y-xh}$
  3. $\displaystyle \frac{2hy}{y+xh}$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

area of 1st parallelogram = y cm square, height = h cm

Area= Base\times Height
$y=b\times h$ ----eq 1
Base =$b=y/h$
area of 2nd parallelogram = 2y cm square, height = H cm
Base of 2nd parallelogram = $\dfrac { y }{ h } +x$
$2y=(\dfrac { y }{ h }+x)\times H$
$H=\dfrac { 2yh }{ (y+hx) } $

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

The base of a parallelogram is three times its height. If the area of the parallelogram is $75$ sq cm, then its height is

  1. $5 cm$
  2. $\displaystyle 5\sqrt{2}cm$
  3. $\displaystyle 3\sqrt{2}cm$
  4. $15 cm$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

let height of ||g be 'b', then base of ||g be 3b.
 area of ||g$=$base$\times $height
$75c{ m }^{ 2 }=3b\times b\ { b }^{ 2 }=25$
 $b=5$cm, therefore height $=5$cm

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

A triangle and a parallelogram are constructed on the same base such that their areas are equal. If the altitude of the parallelogram is $100 m $, then the altitude of the triangle is:

  1. $100 m$
  2. $200 m$
  3. $100\sqrt{2}$ m
  4. $10\sqrt{2}$ m
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the altitude of the $ \Delta =h _{1}$ and altitude of the parallelogram $  =h _{2}$ 

Let base of both $\Delta $ and parallelogram$ = b$
Then, $ b\times h _{2}=\cfrac{1}{2}\times b\times h _{1}$ 
$ \Rightarrow h _{1}=2h _{2}$
$\Rightarrow h _1 = 2\times 100m=200m$

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

The ratio of two adjacent sides of a parallelogram is $3 : 4$ Its perimeter is $105$ cm Find its area if altitude corresponding to the larger is $15$ cm

  1. $900$ $\displaystyle cm^{2}$
  2. $600$ $\displaystyle cm^{2}$
  3. $300$ $\displaystyle cm^{2}$
  4. $450$ $\displaystyle cm^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the two adjacent sides of the parallelogram be $3x$ and $4x$ Then
$2(3x + 4x) = 105$
$\displaystyle \Rightarrow 14x=105.$
$\displaystyle \Rightarrow x=7.5$
$\displaystyle \therefore $ The two sides are $3 \times 7.5$ cm and $4 \times 7.5$ cm 
i.e, $22.5$ cm and $30$ cm
Area of the parallelogram = base x altitude 
                                         = $30$ cm $\times$ $15$cm = $450$$\displaystyle cm^{2}$

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

ABCD is a parallelogram P and R are two points on AB such that the area of parallelogram ABCD is 8 times the are of $\displaystyle \Delta DPR$ If PR = 5cm then CD is equal to 

  1. $10$ cm
  2. $5$ cm
  3. $20$ cm
  4. $12$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Area of parallelogram ABCD 
= $8$ X Area of $\displaystyle \Delta $DPR
$\displaystyle \Rightarrow AB\times \ height=8\times \left ( \frac{1}{2}\times PR\times height \right )$
(Note: Height will be same for both the $\displaystyle \Delta $ and parallelogram)
$\displaystyle \Rightarrow  $ $AB = 4\times  PR = 4$ $\displaystyle \times  5$ cm = $20$ cm
$\displaystyle \therefore $ $\displaystyle CD = 20 $ cm

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

If the base of a parallelogram is $(x + 4)$, altitude to the base is $(x - 3)$ and the area is $ \displaystyle \left ( x^{2}-4 \right )$, then what is the actual area equal to?

  1. $60$ sq units
  2. $45$ sq units
  3. $77$ sq units
  4. $96$ sq units
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Area of the parallelogram $=$ base $\times$ altitude 
$ \Rightarrow \left ( x+4 \right )\times\left ( x-3 \right )=x^{2}+4x-3x-12$
$ \displaystyle=x^{2}+x-12$
Given, $ \displaystyle x^{2}+x-12=x^{2}-4$

$\Rightarrow x=8$
$\displaystyle \therefore $Actual area $\displaystyle = \left ( 8 \right )^{2}-4 = 64 - 4 = 60$ sq units

Multiple choice maths perimeter and area of rectilinear figures parallelogram and rectangle area of parallelogram area of a parallelogram

A parallelogram whose sides are 10 cm and 5 cm has one diagonal of 8 cm, then the length of the other diagonal is

  1. 12 cm

  2. 11 cm

  3. 14 cm

  4. none of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given that, a parallelogram whose sides$l=10cm$ and$b=5cm$ has one diagonal${{d} _{1}}=8cm$.

Let, length of other diagonal $={{d} _{2}}$.


Now we know that,

  $ {{d} _{1}}^{2}+{{d} _{2}}^{2}=2\left( {{l}^{2}}+{{b}^{2}} \right) $

 $ {{8}^{2}}+{{d} _{2}}^{2}=2\left( {{10}^{2}}+{{5}^{2}} \right) $

 $ {{d} _{2}}^{2}=250-64 $

 $ {{d} _{2}}^{2}=186 $

 $ {{d} _{2}}=\sqrt{186}cm $


Hence, this is the answer.