Tag: maths

Questions Related to maths

In a kite shaped figure $ ABCD $, $ AB= AD $  and $ CB= CD $. Points $ P, Q $ and $ R $ are mid-points of sides $ AB, BC $ and $ CD $ respectively. Then, find $ \angle PQR  $
maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

  1. $45^{\circ}$

  2. $30^{\circ}$

  3. $90^{\circ}$

  4. $60^{\circ}$

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Correct Option: C
Explanation:

Given: ABCD is a kite shaped figure. $AB = AD$, $BC = CD$, P, Q< R are mid points of AB, BC and CD respectively


To Prove: $\angle PQR = 90^{\circ}$

Construction: Join AC and BD

In $\triangle ABC$

P is mid point of AB and Q is mid point of BC

Then, $PQ \parallel AC$ (By Mid point theorem)

In $\triangle BCD$

Q is mid point of BC and R is mid point of CD

Then, $QR \parallel BD$ (mid point theorem)

We know, diagonals of the Kite shaped figure, bisect at right angles.
thus, $AC \perp BD$ 

hence, $PQ \perp QR$ (Angle made between two lines is same as the angle between their corresponding parallel sides)
or $\angle PQR = 90^{\circ}$

State whether True or False:
All kites are rhombuses

maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

  1. True

  2. False

  3. Ambiguous

  4. Data insufficient

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Correct Option: B
Explanation:
Since all kites do not have equal sides.
Hence, the answer is the statement is false.

If angles P, Q, R and S of the quadrilateral PQRS taken in order are in the ratio 3 : 7 : 6 : 4 then PQRS is a

maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

  1. rhombus

  2. parallelogram

  3. trapezium

  4. kite

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Correct Option: C
Explanation:

$\Rightarrow$  $\angle P:\angle Q:\angle R:\angle S=3:7:6:4$    [ Given ]

$\Rightarrow$  Let $\angle P=3x,\,\angle Q=7x\,\angle R=6x$ and $\angle S=4x$.
$\Rightarrow$  In quadrilateral PQRS,
$\Rightarrow$  $\angle P+\angle Q+\angle R+\angle S=360^o$
$\therefore$   $3x+7x+6x+4x=360^o$
$\therefore$   $20x=360^o$
$\therefore$   $x=18^o$
$\Rightarrow$  $\angle P=3\times 18^o=54^o$
$\Rightarrow$  $\angle Q=7\times 18^o=126^o$
$\Rightarrow$  $\angle R=6\times 18^o=108^o$
$\Rightarrow$  $\angle S=4\times 18^o=72^o$
$\Rightarrow$  Now, $\angle P+\angle Q=54^o+126^o=180^o$
$\Rightarrow$  and $\angle R+\angle S=180^o+ 72^o=180^o$
$\Rightarrow$  If transversal intersects two lines in such a way that a pair of consecutive interior angles are supplementary, then the two lines are parallel.
$\therefore$    $PS\parallel QR$
$\therefore$  Quadrilateral $PQRS$ is a trapezium.

The area of a trapezium is $24{ cm }^{ 2 }$. The distance between its parallel sides is $4 cm$. If one of the parallel sides is $7 cm$. What is the measure of the other parallel side ?

maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

  1. $5 cm$

  2. $8 cm$

  3. $12 cm$

  4. $7 cm$

Show answer
Correct Option: A
Explanation:

Given that,


The area of the trapezium $A = 24c{m^2}$
Height $h = 4cm$

One of the side $a = 7cm$
Let the other side be $x$

SInce,
$A = \dfrac{1}{2} \times \left( {a + x} \right) \times h$
$24 = \dfrac{1}{2} \times \left( {7 + x} \right) \times 4$

$\dfrac{{24}}{2} = 7 + x$

$12 - 7 = x$

$x = 5$

Hence , the other side be the $5 \,cm$

If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification:

maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

  1. rhombus

  2. rectangle

  3. square

  4. isosceles trapezoid

  5. none of these

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Correct Option: E
Explanation:

Diagonals of a square, rhombus are perpendicular but a quadrilateral with perpendicular diagonal need not be a square, rhombus. 

ABCD is a kite in which AB=AD and CB=CD, if $\angle ABD=40^{0},$ $then find \angle A+\angle C.$

maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

  1. $220^{0}$

  2. $200^{0}$

  3. $180^{0}$

  4. $210^{0}$

Show answer
Correct Option: B
Explanation:

we have,

$ABCD$ is a kite.

Then, Given that,

$ AB=AD\,\,\,\,\,.......\,\,\left( 1 \right) $

$ CB=CD\,\,\,\,\,.......\,\,\left( 2 \right) $

If $\angle ABD={{40}^{o}}$

Then, show Diagram,

$\angle ABD=\angle ADB={{40}^{o}}$

And

$ \angle ADB=\angle CBD={{40}^{o}}\,\,\left( \text{Alternate}\,\text{angle} \right) $

$ \angle ABD=\angle CDB={{40}^{o}}\,\,\,\left( \text{Alternate}\,\text{angle} \right) $

Then $\angle B+\angle D={{40}^{o}}+{{40}^{o}}+{{40}^{o}}+{{40}^{o}}={{160}^{o}}$

But, We know that,

In any quadrilateral

$ \angle A+\angle B+\angle C+\angle D={{360}^{o}} $

$ \angle A+\angle C={{360}^{o}}-{{160}^{o}} $

$ \angle A+\angle C={{200}^{o}} $

Hence, this is the answer.

$ABCD$ is trapezium with $AB||DC=30cm$ and $AB=50cm$. If $X$ and $Y$ are the mid point of $AD$ and $BC$, then $ar(DCYX)=\dfrac {5}{9}ar(XYBA)$.

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. True

  2. False

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Correct Option: B

The length of the parallel sides of a trapezium are 14 cm and 7 cm. If the length of third side is 8 cm and of fourth sides is c xm, then the number of possible integral value of x is :

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. 12

  2. 13

  3. 14

  4. 17

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Correct Option: A

A quadrilateral having one and only one pair of parallel sides is called 

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. a parallelogram

  2. a kite

  3. a rhombus

  4. trapezoids

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Correct Option: D
Explanation:

Quadrilateral with one pair of parallel sides is a trapezoid. If it has two pairs of parallel sides it is a parallelogram, but parallelograms are also trapezoids.

State 'true' or 'false'

The diagonals of a trapezium bisect each other.

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

False.
Diagonals of trapezium does not bisect each other.