Tag: constructing a perpendicular bisector

Questions Related to constructing a perpendicular bisector

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.

A vertex of square is $(3,4)$ and diagonal's equation is given by $x+2y=1$,then the second diagonal which passes through given vertex will be 

maths drawing of different geometrical figures constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector

  1. $2x-y+2=0$

  2. $x+2y=11$

  3. $2x-y=2$

  4. None of these

Show answer
Correct Option: C
Explanation:
Diagonals of Square are perpendicular bisector of each other.

$x+2y=1 \Rightarrow y=\dfrac{-x}{2}+\dfrac{1}{2}$

Comparing with $y=mx+c$ we have $slope=m _1=\dfrac{-1}{2}$

As diagonals are perpendicular, so

 $m _1\times m _2=-1$

$\dfrac{-1}{2}\times m _2=-1$

$m _2=2$

equation of other diagonal is $y=2x+c$

as this passes through $(3,4)$ it satifies the equation

$\Rightarrow 4=2(3)+c \Rightarrow c=-2$

Thus equation of diagonal $\equiv 2x-y-2=0$

Hence, the answer is option $C$

With ruler and compasses,we can bisect any given line segment.

maths drawing of different geometrical figures constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. True

  2. False

Show answer
Correct Option: A

$A B C$  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

maths geometrical construction constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $120 ^ { \circ }$

  2. $180 ^ { \circ }$

  3. $60 ^ { \circ }$

  4. $150 ^ { \circ }$

Show answer
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

maths geometrical construction constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. 15, 6

  2. 33, 12

  3. 5, 0

  4. 14, 6

  5. none of these

Show answer
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.