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Basic constructions

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$\overset \leftrightarrow{PQ}$ is perpendicular to $\overset \leftrightarrow{RS}$ is symbolically written as

  1. $\overset \leftrightarrow{PQ}\, \perp \, \overset \leftrightarrow{RS}$

  2. $\overset \leftrightarrow{PQ}\, \parallel \, \overset \leftrightarrow{RS}$

  3. $\overset \leftrightarrow{PQ}\, \neq \, \overset \leftrightarrow{RS}$

  4. $\overset \leftrightarrow{PQ}\, = \, \overset \leftrightarrow{RS}$


Correct Option: A
Explanation:

$ \overleftrightarrow { PQ } $ is perpendicular to $ \overleftrightarrow { RS } $ is symbolically written as $ \overleftrightarrow { PQ } \bot  \overleftrightarrow { RS }  $

When two line segments meet at a point forming right angle they are said to be __________ to each other.

  1. Parallel

  2. Perpendicular

  3. Equal

  4. None of the above


Correct Option: B
Explanation:

When two line segments meet at a point forming right angle they are said to be perpendicular to each other.

$\displaystyle \overleftrightarrow {PQ}$ is perpendicular to $\displaystyle \overleftrightarrow {RS}$ is symbolically written as:

  1. $\displaystyle \overleftrightarrow {PQ}\perp \overleftrightarrow {RS}$

  2. $\displaystyle \overleftrightarrow {PQ}\parallel \overleftrightarrow{RS}$

  3. $\displaystyle \overleftrightarrow {PQ}\neq \overleftrightarrow{RS}$

  4. $\displaystyle \overleftrightarrow{PQ}= \overleftrightarrow {RS}$


Correct Option: A
Explanation:

$ \overleftrightarrow { PQ } $ is perpendicular to $ \overleftrightarrow { RS } $ is symbolically written as $\overleftrightarrow { PQ } \bot  \overleftrightarrow { RS } $

When two lines are perpendicular to each other, the angle is said to be _______ angle.

  1. acute

  2. right

  3. obtuse

  4. equal


Correct Option: B
Explanation:

Two given lines are perpendicular means the angle between them is $90^o$, i.e. a right angle.

When a perpendicular is drawn to a given line, in what ratio is the line divided into?

  1. $1:1$

  2. $1:2$

  3. $2:1$

  4. Cannot be said


Correct Option: D
Explanation:

A line does not have a definite length.

Hence, when a perpendicular is drawn to the given line, nothing can be said about the ratio it gets divided into.

The steps to construct a line perpendicular to $XY$ and passing through $P$ is given in random order :
$1.$ Move the set square along XY so the other short side touches Point P.
$2.$ Use the edge of the set square to draw a line through Point P.
$3.$ Draw a line $XY$ and mark point $P$.
$4.$ Place one short side of the set square on the line XY.
Which of the following will be the fourth step :

  1. $1$

  2. $2$

  3. $3$

  4. $4$


Correct Option: B
Explanation:

$1.$ Draw a line $XY$ and mark a point $P$ on it.

$2.$ Place one short side of the set square on the line $XY$.
$3.$ Move the set square along $XY$ so the other short side touches point $P$.
$4.$ Use the edge of the set square to draw a line through point $P$ .
So $2.$ is the fourth step.
Option $B$ is correct.

The steps to construct a line perpendicular to $XY$ and passing through $P$ is given in random order :
$1.$ Move the set square along XY so the other short side touches Point P.
$2.$ Use the edge of the set square to draw a line through Point P.
$3.$ Draw a line $XY$ and mark point $P$.
$4.$ Place one short side of the set square on the line XY.
Which of the following will be the first step :

  1. $1$

  2. $2$

  3. $3$

  4. $4$


Correct Option: C
Explanation:

$1.$ Draw a line $XY$ and mark a point $P$ on it.

$2.$ Place one short side of the set square on the line $XY$.
$3.$ Move the set square along $XY$ so the other short side touches point $P$.
$4.$ Use the edge of the set square to draw a line through point $P$ .
So $3.$ is the first step.
Option $C$ is correct.

The steps to construct a line perpendicular to $XY$ and passing through $P$ is given in random order :
$1.$ Move the set square along XY so the other short side touches Point P.
$2.$ Use the edge of the set square to draw a line through Point P.
$3.$ Draw a line $XY$ and mark point $P$.
$4.$ Place one short side of the set square on the line XY.
Which of the following will be the second step :

  1. $1$

  2. $2$

  3. $3$

  4. $4$


Correct Option: D
Explanation:

$1.$ Draw a line $XY$ and mark a point $P$ on it.

$2.$ Place one short side of the set square on the line $XY$.
$3.$ Move the set square along $XY$ so the other short side touches point $P$.
$4.$ Use the edge of the set square to draw a line through point $P$ .
So $4.$ is the second step.
Option $D$ is correct.

To construct a perpendicular to a line ($L$) from a point ($P$) outside the line, steps are given in jumbled form.Identify the first step from the following.
1) Draw line $PQ$.
2)Draw a line $L$ and consider point $P$ outside the line.
3)Take $P$ as a center, draw $2$ arcs on line $L$ and name it as points $A$ and $B$ respectively.
4)Taking $A$ and $B$ as a center one by one and keeping the same distance in compass, draw the arcs on other side of the plane.The point where these arcs intersect name that point as $Q$.

  1. $4$

  2. $3$

  3. $2$

  4. $1$


Correct Option: C
Explanation:

The correct sequence is:

Step 1. Draw a line $L$ and consider a point $P$ outside the line.
Step 2. Take $P$ as center and draw two arcs on line $L$ ans name the points $A$ and $B$ respectively.
Step 3.Taking $A$ and $B$ as centres one by one and keeping the same distance in compass , draw the arcs on other side of the plane .The point where these arcs intersect name that as $Q$
Step 4. Draw line $PQ$
So the first step is $2$
Option $C$ is correct.

When a perpendicular is drawn to a given line and it also bisects it, then the perpendicular divides the line into

  1. $1:1$

  2. $1:2$

  3. $2:3$

  4. None of the above


Correct Option: A
Explanation:

Bisects means division into two equal parts .

When a perpendicular is drawn to a given line and it also bisects it, then the perpendicular divides the line into
Thus the correct answer is $1: 1$

The steps to construct a line perpendicular to $XY$ and passing through $P$ is given in random order :
$1.$ Move the set square along XY so the other short side touches Point P.
$2.$ Use the edge of the set square to draw a line through Point P.
$3.$ Draw a line $XY$ and mark point $P$.
$4.$ Place one short side of the set square on the line XY.
Which of the following will be the third step :

  1. $1$

  2. $2$

  3. $3$

  4. $4$


Correct Option: A
Explanation:

$1.$ Draw a line $XY$ and mark a point $P$ on it.

$2.$ Place one short side of the set square on the line $XY$.
$3.$ Move the set square along $XY$ so the other short side touches point $P$.
$4.$ Use the edge of the set square to draw a line through point $P$ .
So $1.$ is the third step.
Option $A$ is correct.

To construct a perpendicular to a line ($L$) from a point ($P$) outside the line, steps are given in jumbled form.Identify the third step from the following.
1) Draw line $PQ$.
2)Draw a line $L$ and consider point $P$ outside the line.
3)Take $P$ as a center, draw $2$ arcs on line $L$ and name it as points $A$ and $B$ respectively.
4)Taking $A$ and $B$ as a center one by one and keeping the same distance in compass, draw the arcs on other side of the plane.The point where these arcs intersect name that point as $Q$.

  1. $4$

  2. $3$

  3. $2$

  4. $1$


Correct Option: A
Explanation:

The correct sequence is:

Step 1. Draw a line $L$ and consider a point $P$ outside the line.
Step 2. Take $P$ as center and draw two arcs on line $L$ ans name the points $A$ and $B$ respectively.
Step 3.Taking $A$ and $B$ as centres one by one and keeping the same distance in compass , draw the arcs on other side of the plane .The point where these arcs intersect name that as $Q$
Step 4. Draw line $PQ$
So the third step is $4$
Option $A$ is correct.

To construct a perpendicular to a line ($L$) from a point ($P$) outside the line, steps are given in jumbled form.Identify the second step from the following.
1)Draw line $PQ$.
2)Draw a line $L$ and consider point $P$ outside the line.
3)Take $P$ as a center, draw $2$ arcs on line $L$ and name it as points $A$ and $B$ respectively.
4)Taking $A$ and $B$ as a center one by one and keeping the same distance in compass, draw the arcs on other side of the plane.The point where these arcs intersect name that point as $Q$.

  1. $4$

  2. $3$

  3. $2$

  4. $1$


Correct Option: B
Explanation:

The correct sequence is:

Step 1. Draw a line $L$ and consider a point $P$ outside the line.
Step 2. Take $P$ as center and draw two arcs on line $L$ ans name the points $A$ and $B$ respectively.
Step 3.Taking $A$ and $B$ as centres one by one and keeping the same distance in compass , draw the arcs on other side of the plane .The point where these arcs intersect name that as $Q$
Step 4. Draw line $PQ$
So the second step is $3$
Option $B$ is correct.

There is a rectangular sheet of dimension $(2m-1)\times (2n-1)$, (where $m > 0, n > 0$). It has been divided into square of unit area by drawing lines perpendicular to the sides. Find number of rectangles having sides of odd unit length?

  1. $(m+n+1)^2$

  2. $mn(m+1)(n+1)$

  3. $4^{m+n-2}$

  4. $m^2n^2$


Correct Option: D
Explanation:

Total no. of horizontal line=2m

Total no. of vertical lines=2n
($\because$ Each line is at unit distance and hence, total no. of lines=Distance/lenght +1).
To form a square from three lines,we must select one even and one odd numbered horizontal and vertical line.
$\therefore$ Ways possible of selecting such squares=$({ C } _{ 1 }^{ m }\times { C } _{ 1 }^{ m })\times ({ C } _{ 1 }^{ n }\times { C } _{ 1 }^{ n })$
$ ={ C } _{ 1 }^{ m }\times { C } _{ 1 }^{ m }\times { C } _{ 1 }^{ n }\times { C } _{ 1 }^{ n }$
$ ={ m }^{ 2 }\times { n }^{ 2 }$
$ ={ m }^{ 2 }{ n }^{ 2 }$

The steps for constructing a perpendicular from point $A$ to line $PQ$ is given in jumbled order as follows: $(A$ does not lie on $PQ)$
1. Join $R-S$ passing through $A$.
2. Place the pointed end of the compass on $A$ and with an arbitrary radius, mark two points $D$ and $E$ on line $PQ$ with the same radius.
3. From points $D$ and $E$, mark two intersecting arcs on either side of $PQ$ and name them $R$ and $S$.
4. Draw a line $PQ$ and take a point $A$ anywhere outside the line.
The second step in the process is:
  1. $1$

  2. $2$

  3. $3$

  4. $4$


Correct Option: B
Explanation:

Correct sequence is :

Step 1 . Draw a line $PQ$ and take a point $A$ anywhere outside the line.

Step 2. Place the pointed end of the compass on $A$ and with an arbitrary radius, mark two points $D$ and $E$ on line $PQ$ with the same radius.

Step 3. From points $D$ and $E$, mark two intersecting arcs on either side of $PQ$ and name them $R$ and $S.$

Step 4. Join $R−S$ passing through $A$.

So the second step is $2$.

Option $B$ is correct.

To construct a perpendicular to a line($L$) from a point ($P$) outside the line, steps are given in jumbled form.Identify the fourth step from the following
1) Draw line $PQ$
2)Draw a line $L$ and consider point $P$ outside the line
3)Take P as a center, draw $2$ arcs on line $L$ and name it as points $A$ and $B$ respectively
4)Taking $A$ and $B$ as a center one by one and keeping the same distance in compass, draw the arcs on other side of the line.The point where these arcs intersect name that point as $Q$

  1. $4$

  2. $3$

  3. $2$

  4. $1$


Correct Option: D
Explanation:

The correct sequence is:

Step 1. Draw a line $L$ and consider a point $P$ outside the line.
Step 2. Take $P$ as center and draw two arcs on line $L$ ans name the points $A$ and $B$ respectively.
Step 3.Taking $A$ and $B$ as centres one by one and keeping the same distance in compass , draw the arcs on other side of the line .The point where these arcs intersect name that as $Q$
Step 4. Draw line $PQ$
So the fourth step is $1$
Option $D$ is correct.

$A B C$  is a triangle. The bisectors of the internal angle  $\angle B$  and external angle $\angle C$  intersect at  $D.$  if  $\angle B D C = 60 ^ { \circ }$  then  $\angle A$  is

  1. $120 ^ { \circ }$

  2. $180 ^ { \circ }$

  3. $60 ^ { \circ }$

  4. $150 ^ { \circ }$


Correct Option: C
Explanation:

Consider $\triangle ABC$

Let $BC$ be extended to $E$
Since Angular bisectors Meet at $D$
$\angle ABD=\angle DBC\cdots(1)$
$\angle ACD=\angle DCE\cdots(2)$
Consider $ \triangle DBC$
By External sum property 
$\angle DCE=\angle BDC+\angle DBC$
$\implies 2\angle DCE=2(60^{\circ})+2\angle DBC$
$\implies \angle ACE=120^{\circ}+\angle ABC$
By external sum property of $\triangle ABC$
$\angle ACE=\angle BAC+\angle ABC$
$\implies \angle A=60^{\circ}$

The line segment connecting (x, 6) and (9, y) is bisected by the point (7, 3) Find the values of x and y

  1. 15, 6

  2. 33, 12

  3. 5, 0

  4. 14, 6

  5. none of these


Correct Option: C
Explanation:

Since line segment connecting $(x,6)$ and $(9,y)$ is bisected by the point $(7,3)$


Therefore, $\dfrac {x+9}2=7\Rightarrow x=5$ and $\dfrac {6+y}2=3\Rightarrow y=0$

$\therefore x=5, y=0$

Option C is correct.

If $PQ$ is the perpendicular bisector of $AB$, then $PQ$ divides $AB$ in the ratio:

  1. $1:2$

  2. $1:3$

  3. $2:3$

  4. $1:1$


Correct Option: D
Explanation:

Perpendicular bisector always divides a segment into two equal parts.
Therefore $PQ$ divides $AB$ into $1:1$.

For drawing the perpendicular bisector of $PQ$, which of the following radii can be taken to draw arcs from $P$ and $Q$?

  1. $\dfrac{PQ}2$

  2. $\dfrac{PQ}3$

  3. $\dfrac{2PQ}3$

  4. $\dfrac{PQ}4$


Correct Option: C
Explanation:

To draw a perpendicular bisector of a given side, take any length that is greater than half the length of the side. Draw the arcs from the edges of the base. The point where arcs meet is on the perpendicular bisector.


From the given options,

$\dfrac{2PQ}{3}$ can be considered to draw to draw arcs from edges $P, \ Q$

Remaining options has the value $\leq \dfrac{PQ}{2}$

The instrument in the geometry box having the shape of a triangle is called a 

  1. Protractor

  2. Compasses

  3. Divider

  4. Set-square


Correct Option: D
Explanation:

The instrument in the geometry box having the shape of a triangle are called set-squares

Two parallel lines have _____ slopes.

  1. opposite

  2. equal

  3. negative

  4. different


Correct Option: B
Explanation:

Two parallel lines have equal slopes as they are at the same inclination with the positive direction of X-axis. Also, the coefficients of $x$ and $y$ of two parallel lines are in proportion.

Option $B$ is correct.

In the sides a,b,c of a triangle ABC are in A.P then $\dfrac{b}{c}$ belong to

  1. $(0, \dfrac{2}{3})$

  2. $(1,2)$

  3. $(\dfrac{2}{3}, 2)$

  4. $(\dfrac{2}{3}, \dfrac{7}{3})$


Correct Option: A

With compasses and ruler, construct with each of the following angles:

  1. 60 $ ^{\circ} $

  2. 30 $ ^{\circ} $

  3. 90 $ ^{\circ} $

  4. 45 $ ^{\circ} $

  5. 22 $\frac{1}{2} ^{\circ} $

  6. 75 $ ^{\circ} $


Correct Option: A
State True or False:
An angle of $52.5$ can be constructed using the compass.
  1. True

  2. False


Correct Option: A
Explanation:

 As $52.5 =\frac{210^{\circ}}{4} \ and \   210 =180 + 30 $  and we can bisect the angle, so given angle can be constructed.

So True.

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