Quadrilaterals Test

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Quadrilaterals Test - 31

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Weekly Quiz Competition

Question 1
1 / -0

In the adjoining figure, if $$\angle A =90^o$$, then find $$x$$.

A
$$11$$

B
$$19$$

C
$$13$$

D
$$20$$

Solution

We know, by angle sum property, the sum of angles of a quadrilateral is $$360^o$$.

The given angles are $$\angle A=90^o, \angle B=6x-5^o, \angle C=7x-15^o, \angle D=2x+5^o$$.

Then, $$\angle A+\angle B+\angle C+\angle D=360^o$$

$$\implies$$ $$90^o+6x-5^o+7x-15^o+2x+5^o=360^o$$
$$\implies$$ $$15x+75^o=360^o$$
$$\implies$$ $$15x=360^o-75^o=285^o$$
$$\implies$$ $$x=19^o$$.

Therefore, option $$B$$ is correct.
Question 2
1 / -0

The angles of a quadrilateral are in the ratio $$1:2:3:4$$. The largest angle is:

A
$$36^{o}$$

B
$$72^{o}$$

C
$$108^{o}$$

D
$$144^{o}$$

Solution

Given ratio of angles of quadrilateral $$ABCD$$ is $$1:2:3:4$$.
Let the angles of quadrilateral $$ABCD$$ be $$1x,2x,3x,4x$$, respectively.

We know, by angle sum property, the sum of angles of a quadrilateral is $$360^o$$.
$$\Rightarrow 1x+2x+3x+4x=360^o$$
$$\Rightarrow 10x=360^o$$
$$\therefore x=36^o$$.

$$\therefore \angle A =1x=1 \times 36^o =36^o$$,
$$ \angle B =2x=2\times 36^o =72^o$$,
$$ \angle C =3x=3 \times 36^o =108^o$$
and $$ \angle D =4x=4\times 36^o =144^o$$.

$$\therefore$$ The largest angle $$=144^o$$.

Hence, option $$D$$ is correct.
Question 3
1 / -0

In a quadrilateral $$ABCD$$, $$\angle{A}+\angle{C}$$ is $$2$$ times of $$\angle{B}+\angle{D}$$. If $$\angle {D}={40}^{o}$$, then $$\angle {B}=$$ _____.

A
$${60}^{o}$$

B
$${80}^{o}$$

C
$${100}^{o}$$

D
$${120}^{o}$$

Solution

Given, $$\angle A+\angle C=2\left(\angle B+\angle D\right)$$.
Also, by angle sum property,
$$\angle A+\angle B+\angle C+\angle D={360}^{0}$$
$$\implies$$ $$\angle A+\angle C+\angle B+\angle D={360}^{0}$$
$$\implies$$ $$2\left(\angle B+\angle D \right)+ \left(\angle B+\angle D \right)= {360}^{0}$$
$$\implies$$ $$ 3\left(\angle B+\angle D \right)= {360}^{0}$$
$$\implies$$ $$\angle B+\angle D={120}^{0}$$
$$\implies$$ $$\angle B+{40}^{0}={120}^{0} $$
$$\implies$$ $$ \angle B={80}^{0}$$.

Therefore, option $$B$$ is correct.
Question 4
1 / -0

Read the following statement and choose the correct alternative from those given below them:
(i) Diagonals of a rectangle are perpendicular bisectors of each other
(ii) Diagonals of a rhombus are perpendicular bisectors of each other
(iii) Diagonals of a parallelogram are perpendicular bisectors of each other

A
Statement $$(ii)$$ and $$(iii)$$ are true

B
Only statement $$(ii)$$ is true

C
Statement $$(ii)$$ and $$(iv)$$ are true

D
Statement $$(i),(ii)$$ and $$(iv)$$ are true

Solution

$$\Rightarrow$$ Diagonals of a rectangle are equal in length but does not bisect each other perpendicularly. So, first statement is false.
$$\Rightarrow$$ Not all types of parallelogram diagonals bisect each other perpendicularly, so third statement is also false.
$$\Rightarrow$$ Diagonals of rhombus bisect each other perpendicularly.
So, only second statement is correct.
Question 5
1 / -0

The line $$3x+2y=6$$ will divide the quadrilateral formed by the lines $$x+y=5,y-2x=8,3xy+2x=0$$ and $$4y-x=0$$ in

A
two quadrilaterals

B
one pentagon and one triangle

C
two traingles

D
one triangle and one quadrilateral

Solution

$$\begin{array}{l} we\, \, have........... \\ x+y=5\, \, \, \, \, \, (draw\, \, a\, line\, \, on\, \, x-axis\, and\, y-axis\, as\, \, \, (0,5)\, \, \, (5,0) \\ y-2x=8\, \, \, \, (draw\, \, a\, line\, \, on\, \, x-axis\, and\, y-axis\, as\, \, \, (-4,0)\, \, \, (0,8) \\ 3y+2x=0\Rightarrow y=\frac { { 2x } }{ 3 } ,\, \\ 4y-x=0\Rightarrow y=\frac { x }{ 4 } ,\, \, \, \, \end{array}$$
we get the quadrilateral point ABCD,
and we find out the $$3x + 2y = 6$$
$$\begin{array}{l} 3x+2y=6-----(i) \\ equ\, \, (i)\, \, divide\, by\, \, 6 \\ \frac { x }{ 2 } +\frac { y }{ 3 } =1\, \, \, then,\, we\, get\, two\, quadrilateral\, \, as\, (AFED,EFBC) \\ So,\, that\, the\, \, \, correct\, option\, is\, A. \end{array}$$
Question 6
1 / -0

$$A B C D$$ is a quadrilateral. If $$\angle A C B = \angle A D B ,$$ then $$A B C D$$ is

A
A cyclic quadrilateral

B
A parallelogram

C
A square

D
Not a cyclic quadrilateral

Solution

If $$\angle ACB=\angle ADB$$,

Then the angles in the same segment are equal.

i.e., $$A, B, C, D$$ are concyclic.

Hence ABCD is a cyclic quadrilateral.
Question 7
1 / -0

X, Y are the mid-points of opposite sides AB and DC respectively of a parallelogram ABCD. AY and DX are intersecting at S; CX and BY are intersecting at R. Then SXRY is a_____________.

A
Rectangle

B
Kite

C
Parallelogram

D
Square

Solution

Given: $$X$$ and $$Y$$ are the mid-points of opposite sides $$AB$$ and $$DC$$ of a parallelogram $$ABCD.$$

$$AX = XB$$ and $$DY = YC$$

$$AB = DC$$

$$XB = DY......(i)$$

Also $$AB || DC...... $$(opp. sides of a ïïgm)
$$XB || DY.....(ii)$$

Since in quadrilateral $$XBYD, XB = DY$$ and $$XB || DY$$

$$SYRX$$ is a parallelogram.
Question 8
1 / -0

If the angles of quadrilateral are $$x,x+20,2x+45,x+15$$. Find $$x$$.

A
$$52$$

B
$$56$$

C
$$58$$

D
$$60$$

Solution

Given: The angles of quadrilateral are $$x,x+20,2x+45,x+15$$

By angle sum property, the sum of all the angles of a quadrilateral is $$360^\circ$$

$$x+(x+20^\circ)+(2x+45^\circ)+(x+15^\circ)=360^\circ$$

$$\implies$$ $$5x+80^\circ=360^\circ$$

$$\implies$$ $$5x=360^\circ-80^\circ$$

$$\implies$$ $$5x=280^\circ$$

$$\implies$$ $$x=56^\circ$$

Therefore, option $$B$$ is correct.
Question 9
1 / -0

In a quadrilateral $$ABCD,\angle A = 2x - {35^ \circ },\angle B = 3x - {5^ \circ },\angle C = x + {10^ \circ },\angle D = 4x + {20^ \circ },$$ find the value of $$x$$,

A
$$35$$

B
$$37$$

C
$$39$$

D
None of these
Question 10
1 / -0

The angles of quadrilateral are $$30,150,60$$ find the third angle

A
$$120$$

B
$$200$$

C
$$140$$

D
$$50$$

Solution

The angles of quadiateral are $$30,150,60$$
The sum of angles is $$360$$
$$\implies x+30+150+60=360\\x=360-240\\x=120$$

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Quadrilaterals Test - 31

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Quadrilaterals Test - 31

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SHARING IS CARING

Question 1

$$11$$

$$19$$

$$13$$

$$20$$

Solution

Question 2

$$36^{o}$$

$$72^{o}$$

$$108^{o}$$

$$144^{o}$$

Solution

Question 3

$${60}^{o}$$

$${80}^{o}$$

$${100}^{o}$$

$${120}^{o}$$

Solution

Question 4

Statement $$(ii)$$ and $$(iii)$$ are true

Only statement $$(ii)$$ is true

Statement $$(ii)$$ and $$(iv)$$ are true

Statement $$(i),(ii)$$ and $$(iv)$$ are true

Solution

Question 5

two quadrilaterals

one pentagon and one triangle

two traingles

one triangle and one quadrilateral

Solution

Question 6

A cyclic quadrilateral

A parallelogram

A square

Not a cyclic quadrilateral

Solution

Question 7

Rectangle

Kite

Parallelogram

Square

Solution

Question 8

$$52$$

$$56$$

$$58$$

$$60$$

Solution

Question 9

$$35$$

$$37$$

$$39$$

None of these

Question 10

$$120$$

$$200$$

$$140$$

$$50$$

Solution

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