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Construction of perpendicular - class-VIII

Description: construction of perpendicular
Number of Questions: 17
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Tags: basic geometrical concepts and shapes lines and angles geometrical construction practical geometry equal triangles plane geometry line and angle drawing of different geometrical figures shapes and geometric reasoning perpendicular lines maths properties of angles and lines
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$\overset \leftrightarrow{PQ}$ is perpendicular to $\overset \leftrightarrow{RS}$ is symbolically written as

maths drawing of different geometrical figures constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

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

Show answer
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.

maths drawing of different geometrical figures constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. Parallel

  2. Perpendicular

  3. Equal

  4. None of the above

Show answer
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:

maths drawing of different geometrical figures constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

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

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

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

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

Show answer
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.

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. acute

  2. right

  3. obtuse

  4. equal

Show answer
Correct Option: B
Explanation:

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

A perpendicular is drawn using

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line

  1. scale

  2. scale protractor

  3. set square

  4. divider

Show answer
Correct Option: B
Explanation:

A perpendicular is drawn using scale, protractor as well as set squares.

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

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $1:1$

  2. $1:2$

  3. $2:1$

  4. Cannot be said

Show answer
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 :

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
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 :

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
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 :

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
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 $Q$.

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $4$

  2. $3$

  3. $2$

  4. $1$

Show answer
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

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $1:1$

  2. $1:2$

  3. $2:3$

  4. None of the above

Show answer
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 :

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
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 $Q$.

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $4$

  2. $3$

  3. $2$

  4. $1$

Show answer
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 $Q$.

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $4$

  2. $3$

  3. $2$

  4. $1$

Show answer
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?

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

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

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

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

  4. $m^2n^2$

Show answer
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:
maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
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 $Q$

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $4$

  2. $3$

  3. $2$

  4. $1$

Show answer
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.

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