Class 10 Exam > Class 10 Videos > 01 - How to construct a tangent from a point outside the circle? - Class 10 - Maths
01 - How to construct a tangent from a point outside the circle? - Class 10 - Maths Video Lecture
FAQs on 01 - How to construct a tangent from a point outside the circle? - Class 10 - Maths Video Lecture
| 1. How do you construct a tangent from a point outside the circle? |  |
Ans. To construct a tangent from a point outside the circle, follow these steps:
1. Draw a circle with its center at the given point.
2. Draw a line segment connecting the center of the circle and the given point.
3. Bisect the line segment to find its midpoint.
4. Draw a perpendicular line to the line segment at its midpoint.
5. The perpendicular line intersects the circle at two points.
6. Draw a line from the given point to one of the points of intersection.
7. This line is the tangent to the circle from the given point.
| 2. What is a tangent line in geometry? | |
Ans. In geometry, a tangent line is a line that touches a curve or a circle at only one point, without crossing through it. This point of contact is called the point of tangency. The tangent line is perpendicular to the radius of the circle at the point of tangency. In the context of circles, a tangent line is often used to determine angles, lengths, and relationships between geometric figures.
| 3. Can a tangent line intersect a circle at two points? | |
Ans. No, a tangent line cannot intersect a circle at two points. By definition, a tangent line touches a circle at only one point, without crossing through it. This point of contact is called the point of tangency. If a line intersects a circle at two points, it is not a tangent line but rather a secant line or a chord of the circle.
| 4. What is the significance of constructing tangents to a circle? | |
Ans. Constructing tangents to a circle is significant in geometry as it helps to determine various properties and relationships between geometric figures. Some of the key significance of constructing tangents to a circle are:
- Tangents are perpendicular to the radius of the circle at the point of tangency, allowing for the calculation of angles and lengths in geometric problems.
- Tangents can be used to find the common tangents of two circles, which are lines that touch both circles without intersecting.
- Tangents are also essential in solving problems involving circles in real-life applications, such as optics, engineering, and physics.
| 5. Are there any other methods to construct a tangent to a circle from a point outside it? | |
Ans. Yes, apart from the method mentioned earlier, there are other methods to construct a tangent to a circle from a point outside it. Some additional methods include:
- Using a compass and straightedge, construct the perpendicular bisector of the line segment connecting the center of the circle and the given point. The point of intersection of the perpendicular bisector and the circle will be the point of tangency.
- If the center of the circle and the given point are connected by a line segment, construct a right-angled triangle with the hypotenuse being the line segment. The altitude drawn from the right angle to the hypotenuse will intersect the circle at the point of tangency.
- Using the concept of the power of a point, calculate the length of the line segment connecting the center of the circle and the given point. Then, using this length, construct a right-angled triangle with the hypotenuse being the line segment. The altitude drawn from the right angle to the hypotenuse will intersect the circle at the point of tangency.
Text Transcript from Video
okay so on my monitor screen right now I
can see that we have a vegetable garden
over here and I can see a house okay so
let us say in this case we have to build
certain ways to get from the house to
the vegetable garden then how can we
construct such ways or roads so let's
say to enter this vegetable garden there
are four gates mainly there is gate
number one in two right in the middle
and gate number three and four right on
top and bottom as such there is no top
and bottom in a circle but let's say if
this is my plane surface and these are
my direction keys so this is going to be
the top of this vegetable garden and
this will be the bottom of this
vegetable garden now if I have to enter
the vegetable garden right from gate
number one then I can build a straight
road from my house to gate number one
and even we can extend this road right
from gate number one and connect to gate
number two as well and what about gate
number three and four okay so if if I
construct a road something like this
from gate number three and another
straight road to gate number four and
then maybe divert those roads towards my
house then yeah I can say that I can
enter the vegetable garden
not just from a straight road but even
from these roads let's name them Road a
then the middle one is road B and the
down one is road C let's say if I
carefully notice my vegetable garden is
actually in a form of a circle correct
and this house that is located somewhere
over here is actually like a point that
is outside the circle
so if you see Road B starts from the
house and it passes through gate number
one and it touches gate number two so I
can say it is like a line segment that
initiates from an external point and
meets the circle at two points so such a
line that intersects the circle at two
points is called a chord and if we
notice Road a and Road see what can we
comment about them we can say unlike
road B
they touch the circle at only one point
and such a line segment that touches the
circle at only one point will be called
a tangent and not a cord so that is the
main difference between a tangent and a
chord a chord touches the circle at two
points and a tangent touches the circle
at only one point
in this particular point where the
tangent touches the circle is called the
point of tangency also a radius is
always perpendicular to the tangent and
by perpendicular what I mean is that the
radius will always make an angle of 90
degrees with the tangent so tangent can
be identified by two things first of all
that it touches the circle at only one
point and secondly that it is
perpendicular to the radius now let us
just change this kind of construction
design a bit I mean let's just keep road
be as it is and Road a and Road C as we
know constructed straight and then they
were diverted towards the house correct
let's just say for a second that I need
a road a and road C to be initiated from
the house just like Road B now after we
get such a construction design what are
the things that we notice first of all
that it has become more convenient to
reach gate number three and gate number
four now why because we don't have to go
like this and then die or towards the
garden but just like Road be even Road a
and Road C will serve like shortcut
roads towards the garden and what is the
other thing that we can notice this gate
number three and gate number four have
to be moved and why is that because now
the point of tangency has been shifted
now before this particular point was the
point of tangency because roads used to
touch this point now in this new
scenario gate number three has to be
constructed like this and gate number
four has to be constructed over here and
this particular path that was there
which was actually like a radius of this
circular garden will have to be now
shifted and we will form a new path now
now if you will notice this particular
path is the radius to this circle
and even in this scenario when we have
shifted these paths and constructed new
gates at new points even then the radius
will always be perpendicular to the
tangent now actually it is very easier
to construct Road a and Road C we just
have to measure this distance between
the house and the garden right but let's
just make it more interesting let us say
we don't even have any kind of measuring
device through to just notice this
distance then what can we do in that
case we will take help of geometry as we
have learnt earlier geometry helps us to
construct any kind of shape very
accurately and before we start this
construction let me just give you the
main idea behind the construction the
idea is if we can somehow make a bigger
circle okay then this bigger circle
should pass to our given circle at two
points and then with the help of a
straightedge we can join the external
point and these two points where the two
circles intersect and then draw two
lines and as a result these two lines
will actually be the tangents that we
are looking for so now let us go step by
step and learn how do we construct such
bigger circle from an external point so
that we get two tangents so step one in
this kind of procedure would be from a
given circle if this is the center then
first we have to join the center to the
external point with the help of a
straightedge
let's say we named our external point as
P and center as o so we have a line
segment P O so the first step in the
construction is to draw a line segment P
O now once we get line segment p uo then
we will go to step two and we will
construct a perpendicular bisector what
is the meaning of perpendicular bisector
well it is hidden in its name
perpendicular obviously means that there
is an angle of ninety degree that is
involved so this line will actually form
an angle of ninety degree with our line
segment P o and if we understand the
term bisector by will mean two and
sector will mean portions it only means
this particular line will divide the
line segment P O into two portions or
two smaller line segments so
perpendicular bisector
will mean a line or a line segment that
is not only 90 degrees but also divides
another line segment into two parts and
how do we make a perpendicular bisector
with the help of a compass so we will
keep the pointed end of the compass on
point O and we will form two arcs like
this and then we will keep the point on
P as the center and we'll cut these two
arcs like this and then with the help of
straightedge we will join these two
points of intersection and the line that
we get is actually a perpendicular
bisector but what can be achieved by
constructing a perpendicular bisector
then we can consider either p.m. or M o
as the radius and we can form another
bigger circle let us see how do we do
that and once we construct a bigger
circle we know that it will intersect
our given circle at two points and now
we know what has to be the next step we
have to join our external points with
these two points of intersection and
when we will do that we will finally get
two tangents and they will be tangent P
a and tangent P B but how do we really
know if it is a tangent well then we
will have to look out for its
characteristics first of all it should
touch our given circle at only one point
so we can see our first tangent touches
the circle at Point a and the second
tangent touches the circle at point B
the second characteristic we learnt is
that this tangent should be
perpendicular to the radius of our given
circle that is it should form an angle
of 90 degrees so how do we prove if this
tangents form an angle of 90 degrees
with the radius of our given circle any
circle can be actually divided into two
semi circles now if you carefully notice
this bigger circle there are actually
two semi circles in this bigger circle
right so let us consider first the upper
semicircle
and do you recollect one very important
property of semicircle yes the property
which says that a diameter of the
semicircle subtends an angle of 90
degree on the circle so if we go to CP o
which is our line segment is actually
the diameter of the bigger circle or the
diameter of the semicircle so if you go
to see this diameter has subtended an
angle of 90 degrees at Point a and if
you notice the other semicircle even
there the diameter subtends an angle of
90 degree at point B so that actually
really explains the reason of making a
perpendicular bisector and forming a
circle which will touch this external
point and also touch the center of the
circle so that's the logic behind
constructing two tangents from an
external point to a given circle if you
understood the concept please hit like
to this video or else post any queries
in the comment section below and for
more such videos do not forget to
subscribe our Channel let's you keep
watching keep learning
can see that we have a vegetable garden
over here and I can see a house okay so
let us say in this case we have to build
certain ways to get from the house to
the vegetable garden then how can we
construct such ways or roads so let's
say to enter this vegetable garden there
are four gates mainly there is gate
number one in two right in the middle
and gate number three and four right on
top and bottom as such there is no top
and bottom in a circle but let's say if
this is my plane surface and these are
my direction keys so this is going to be
the top of this vegetable garden and
this will be the bottom of this
vegetable garden now if I have to enter
the vegetable garden right from gate
number one then I can build a straight
road from my house to gate number one
and even we can extend this road right
from gate number one and connect to gate
number two as well and what about gate
number three and four okay so if if I
construct a road something like this
from gate number three and another
straight road to gate number four and
then maybe divert those roads towards my
house then yeah I can say that I can
enter the vegetable garden
not just from a straight road but even
from these roads let's name them Road a
then the middle one is road B and the
down one is road C let's say if I
carefully notice my vegetable garden is
actually in a form of a circle correct
and this house that is located somewhere
over here is actually like a point that
is outside the circle
so if you see Road B starts from the
house and it passes through gate number
one and it touches gate number two so I
can say it is like a line segment that
initiates from an external point and
meets the circle at two points so such a
line that intersects the circle at two
points is called a chord and if we
notice Road a and Road see what can we
comment about them we can say unlike
road B
they touch the circle at only one point
and such a line segment that touches the
circle at only one point will be called
a tangent and not a cord so that is the
main difference between a tangent and a
chord a chord touches the circle at two
points and a tangent touches the circle
at only one point
in this particular point where the
tangent touches the circle is called the
point of tangency also a radius is
always perpendicular to the tangent and
by perpendicular what I mean is that the
radius will always make an angle of 90
degrees with the tangent so tangent can
be identified by two things first of all
that it touches the circle at only one
point and secondly that it is
perpendicular to the radius now let us
just change this kind of construction
design a bit I mean let's just keep road
be as it is and Road a and Road C as we
know constructed straight and then they
were diverted towards the house correct
let's just say for a second that I need
a road a and road C to be initiated from
the house just like Road B now after we
get such a construction design what are
the things that we notice first of all
that it has become more convenient to
reach gate number three and gate number
four now why because we don't have to go
like this and then die or towards the
garden but just like Road be even Road a
and Road C will serve like shortcut
roads towards the garden and what is the
other thing that we can notice this gate
number three and gate number four have
to be moved and why is that because now
the point of tangency has been shifted
now before this particular point was the
point of tangency because roads used to
touch this point now in this new
scenario gate number three has to be
constructed like this and gate number
four has to be constructed over here and
this particular path that was there
which was actually like a radius of this
circular garden will have to be now
shifted and we will form a new path now
now if you will notice this particular
path is the radius to this circle
and even in this scenario when we have
shifted these paths and constructed new
gates at new points even then the radius
will always be perpendicular to the
tangent now actually it is very easier
to construct Road a and Road C we just
have to measure this distance between
the house and the garden right but let's
just make it more interesting let us say
we don't even have any kind of measuring
device through to just notice this
distance then what can we do in that
case we will take help of geometry as we
have learnt earlier geometry helps us to
construct any kind of shape very
accurately and before we start this
construction let me just give you the
main idea behind the construction the
idea is if we can somehow make a bigger
circle okay then this bigger circle
should pass to our given circle at two
points and then with the help of a
straightedge we can join the external
point and these two points where the two
circles intersect and then draw two
lines and as a result these two lines
will actually be the tangents that we
are looking for so now let us go step by
step and learn how do we construct such
bigger circle from an external point so
that we get two tangents so step one in
this kind of procedure would be from a
given circle if this is the center then
first we have to join the center to the
external point with the help of a
straightedge
let's say we named our external point as
P and center as o so we have a line
segment P O so the first step in the
construction is to draw a line segment P
O now once we get line segment p uo then
we will go to step two and we will
construct a perpendicular bisector what
is the meaning of perpendicular bisector
well it is hidden in its name
perpendicular obviously means that there
is an angle of ninety degree that is
involved so this line will actually form
an angle of ninety degree with our line
segment P o and if we understand the
term bisector by will mean two and
sector will mean portions it only means
this particular line will divide the
line segment P O into two portions or
two smaller line segments so
perpendicular bisector
will mean a line or a line segment that
is not only 90 degrees but also divides
another line segment into two parts and
how do we make a perpendicular bisector
with the help of a compass so we will
keep the pointed end of the compass on
point O and we will form two arcs like
this and then we will keep the point on
P as the center and we'll cut these two
arcs like this and then with the help of
straightedge we will join these two
points of intersection and the line that
we get is actually a perpendicular
bisector but what can be achieved by
constructing a perpendicular bisector
then we can consider either p.m. or M o
as the radius and we can form another
bigger circle let us see how do we do
that and once we construct a bigger
circle we know that it will intersect
our given circle at two points and now
we know what has to be the next step we
have to join our external points with
these two points of intersection and
when we will do that we will finally get
two tangents and they will be tangent P
a and tangent P B but how do we really
know if it is a tangent well then we
will have to look out for its
characteristics first of all it should
touch our given circle at only one point
so we can see our first tangent touches
the circle at Point a and the second
tangent touches the circle at point B
the second characteristic we learnt is
that this tangent should be
perpendicular to the radius of our given
circle that is it should form an angle
of 90 degrees so how do we prove if this
tangents form an angle of 90 degrees
with the radius of our given circle any
circle can be actually divided into two
semi circles now if you carefully notice
this bigger circle there are actually
two semi circles in this bigger circle
right so let us consider first the upper
semicircle
and do you recollect one very important
property of semicircle yes the property
which says that a diameter of the
semicircle subtends an angle of 90
degree on the circle so if we go to CP o
which is our line segment is actually
the diameter of the bigger circle or the
diameter of the semicircle so if you go
to see this diameter has subtended an
angle of 90 degrees at Point a and if
you notice the other semicircle even
there the diameter subtends an angle of
90 degree at point B so that actually
really explains the reason of making a
perpendicular bisector and forming a
circle which will touch this external
point and also touch the center of the
circle so that's the logic behind
constructing two tangents from an
external point to a given circle if you
understood the concept please hit like
to this video or else post any queries
in the comment section below and for
more such videos do not forget to
subscribe our Channel let's you keep
watching keep learning
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