MCQ

MCQ 11 Mark

The lengths of the diagonals of a rhombus are $16\ cm$ and $12\ cm.$ The length of each side of the rhombus is:

A
$8\ cm$
B
$9\ cm$
✓
$10\ cm$
D
$12\ cm$

Answer

Correct option: C.

$10\ cm$

$\text{AO}=\frac{1}{2}\text{AC}=\Big(\frac{1}{2}\times16\Big)=8\ \text{cm}$
$\text{BO}=\frac{1}{2}\text{BD}=\Big(\frac{1}{2}\times12\Big)=6\ \text{cm}$
From the right $\triangle\text{AOB},$ we have,
$\therefore\text{AB}^2=\text{AO}^2+\text{BO}^2$
$\Rightarrow\text{AB}^2=\big\{(8)^2+(6)^2\big\}\text{cm}^2$
$\Rightarrow\text{AB}=\sqrt{100}=10\ \text{cm}$

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MCQ 21 Mark

The two diagonals are not necessarily equal in $a:$

A
Rectangle.
B
Square.
✓
Rhombus.
D
Isosceles trapezium.

Answer

Correct option: C.

Rhombus.

All sides of Rhombus are equal in length but in case of angle it is not necessary to be equal.
If all the angles are equal then it will become a square.
That’s why diagonals of rhombus are not necessary to be equal in length.

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MCQ 31 Mark

If an angle of a parallelogram is two$-$thirds of its adjacent angle, the smallest angle of the parallelogram is:

A
$54^\circ$
✓
$72^\circ$
C
$81^\circ$
D
$108^\circ$

Answer

Correct option: B.

$72^\circ$

Let the measure of the angle be $x^\circ .$
$\therefore\text{x}+\Big(\frac{2}{3}\times\text{x}\Big)=180$
$\Rightarrow\frac{3\text{x}+2\text{x}}{3}=180$
$\Rightarrow5\text{x}=3\times180$
$\Rightarrow\text{x}=\frac{3\times180}{5}=180$
Hence the anlge is $180^\circ .$
Its adjacent $= (180 - 108^\circ ) = 72^\circ .$
Therefore, the smallest angle is $72^\circ .$

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MCQ 41 Mark

The length and breadth of a rectangle are in the ratio $4 : 3.$ If the diagonal measures $25cm$ then the perimeter of the rectangle is:

A
$56\ cm$
B
$60\ cm$
✓
$70\ cm$
D
$80\ cm$

Answer

Correct option: C.

$70\ cm$

Let the length $AB$ be $4x$ and Breadth $BC$ be $3x.$
Each angle of a rectangle is a right angle. We have,
$\therefore \angle\text{ABC}=90^\circ$
From the right $\triangle\text{ABC}:$
$AC^2 = AB^2 + BC^2$
$\Rightarrow (25)^2 = (4x)^2 + (3x)^2$
$\Rightarrow 16x^2 + 9x^2 = 625$
$\Rightarrow 25x^2 = 625$
$\Rightarrow x^2 = 25$
$\Rightarrow x = 5$
Therefore, lenght $= 4 \times 5 = 20\ cm$ and breadth $= 3 \times 5 = 15\ cm.$

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MCQ 51 Mark

Two adjacent angles of a parallelogram are $(2x + 25)^\circ$ and $(3x - 5)^\circ .$ The value of $x$ is:

A
$28$
✓
$32$
C
$36$
D
$42$

Answer

Correct option: B.

$32$

$\therefore (2x + 25) + (3x - 5) = 180$
$\Rightarrow 2x + 25 + 3x - 5 = 180$
$\Rightarrow 5x = 180 - 20$
$\Rightarrow 5x = 160$
$\Rightarrow x = 32$

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MCQ 61 Mark

The diagonals do not necessarily bisect the interior angles at the vertices in a:

✓
Rectangle.
B
Square.
C
Rhombus.
D
All of these.

Answer

Correct option: A.

Rectangle.

In rectangle, only opposite sides are equal which makes diagonals are not to be perpendicular to each other. As diagonals are not perpendicular to each other, they will not bisect the interior angles.

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MCQ 71 Mark

The bisectors of any two adjacent angles of a parallelogram intersect at:

A
$30^\circ$
B
$45^\circ$
C
$60^\circ$
✓
$90^\circ$

Answer

Correct option: D.

$90^\circ$

Let $\ce{ABCD}$ is a parallelogram.

$AE$ and $AD$ is the bisector angles of adjacent angles of $\angle\text{A}$ and $\angle\text{D}.$
As we know that,
$\angle\text{A}+\angle\text{D}=180^\circ ($Sum of interior angles on the same side of traversal is $180^\circ )$
$\frac{1}{2}\angle\text{A}+\frac{1}{2}\angle\text{D}=\frac{1}{2}\times180^\circ$
$=90^\circ\ ...(\text{i})$
Now, in triangle $\ce{AOD},$
$\angle\text{AED}+\frac{1}{2}\angle\text{A}+\frac{1}{2}\angle\text{D}=180^\circ$ $(AE$ and $AD$ is the angle bisector of $\angle\text{A}$ and $\angle\text{D}).$
$\angle\text{AED}+90^\circ=180^\circ ($From $eq (i))$
$\angle\text{AED}=180^\circ-90^\circ=90^\circ$
So, the bisectors of any two adjacent angles of a parallelogram intersect at $90^\circ .$

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MCQ 81 Mark

The diagonals do not necessarily intersect at right angles in a:

✓
Parallelogram.
B
Rectangle.
C
Rhombus.
D
Kite.

Answer

Correct option: A.

Parallelogram.

The diagonals do not necessarily intersect at right angles in a parallelogram. Only opposite sides, opposite angles are equal and diagonal bisects each other in parallelogram. If diagonals intersect each other at right angle then it would be square or rhombus.

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MCQ 91 Mark

If one angle of a parallelogram is $24^\circ$ less than twice the smallest angle then the largest angle of the parallelogram is:

A
$68^\circ$
B
$102^\circ$
✓
$112^\circ$
D
$176^\circ$

Answer

Correct option: C.

$112^\circ$

Let the measure of smallest anlge be $x^\circ$ and other is $(2x - 24)^\circ .$
$\therefore x + (2x - 24) = 180$
$\Rightarrow x + 2x = 180 + 24$
$\Rightarrow 3x = 204$
$\Rightarrow x = 68$
Hence, the samllest angle is $68^\circ .$
Ite adjacent is $= (180 - 68)^\circ = 112^\circ .$
Therefore, the largest angle is $112^\circ .$

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MCQ 101 Mark

In a square $\ce{ABCD}, AB = (2x + 3)cm$ and $BC = (3x - 5)cm.$ Then, the value of $x$ is:

A
$4$
B
$5$
C
$6$
✓
$8$

Answer

Correct option: D.

$8$

We know, all sides are equal of a square. Then,
$\therefore AB = BC$
$\Rightarrow 2x + 3 = 3x - 5$
$\Rightarrow 3x - 2x = 3 + 5$
$\Rightarrow x = 8$

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