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### Angles and Circles

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Video:

Hello all, welcome to our NPTEL Online Certification Courses on Engineering Drawing
and Computer Graphics.
(Refer Slide Time: 00:30)

(Refer Slide Time: 00:34)

We are in module 2 and lecture 8; the module name is Conic Sections. In lecture 7, we
have learned how to bisect a line and an arc; how to draw a perpendicular line; how to
divide a line. In lecture 8, we are covering how to bisect an angle, how to trisect a right
angle, how to divide a circle, how to pass a circle through three points, these are the things
that we are going to learn.
(Refer Slide Time: 01:07)

The first one is how to bisect an angle? So, here, we have an angle AQR; some arbitrary
angle. Now, we want a line that passes through Q such a way that PQC, angle PQC equal
to angle CQR, how to construct that we are going to learn it.
(Refer Slide Time: 01:58)

To bisect an angle PQR, first of all, we have to mark points A; mark point A and B from
Q with an arbitrary radius. So, we have to use Q mark a random radius. So, because we
know from P to Q maybe half of the distance or lower than that, more than that, it does not
matter. But first of all, make an arc that will intersect Q to P at A and Q to R at B.
Then, use points A and B as centers with the same radius or arbitrary radius. Mark a curve
from A as the centre, similarly from B as centre; mark another arc, where it is intersecting
call that point as C. One C is known; from C to Q, join it by a line. Once it is done, this
angle and this angle becomes the same. Let us look at geometric construction on the sheet.
(Refer Slide Time: 03:47)

Let us draw an angle, somewhere mark it. So, call points Q, P, R. Now, with Q as a centre
at some distance with radius mark curves; say these points as A and B. With the same
radius pick B, mark an arc, where it is intersecting, call that point as C. Now, join points

C and Q. If we use this protractor, the ∠PQC will be the same as the ∠CQR. Let us call
these as the alpha angle. This is the way we bisect an angle.
(Refer Slide Time: 05:45)

Let us look at the next one. How to trisect a right angle? For example, we have a right
angle lines AB and AC. This entire thing, we can use protractor directly to divide it.
However, any arbitrary thing, if we would like to construct a similar procedure works.
To trisect it, first of all, we have to pass an arc passing through B and C points with centre
as A and radius as AB because we know BAC is the triangle. What is it making? Angle.
Use A to B as radius, A as the centre, draw a curve arc, that is this arc..

(Refer Slide Time: 07:01)

Once it is done, mark point D from point C. So, C is this point, and D is that point. So, use
an arbitrary radius, mark a point. Similarly, mark point E from point B; mark a point E
with the same radius. Now, join D and A.; similarly, join E and A, after marking.
(Refer Slide Time: 08:02)

Once it is done, we have three portions; I, II, and III parts. So, angle BAD, angle DAE,
angle EAC are all equal. Let us construct that using our geometrical construction. First of
all, we have to draw a perpendicular line, right angle. Use a pencil. Let us mark point A,
point C. We would like to have a perpendicular line. So, I am trying to use a set square
somewhere here; this is the line that we have.

(Refer Slide Time: 09:36)
Now, we have to do the first step a radius A to B passes through that arc; either A to B or
A to C. So, let us tighten that, draw an arc. Once it is done, extend through scale so that
we have point B also here and an arc passing through B and C. From C, with the same
radius, mark an arc; similarly, from B mark an arc, join. The points are D;, the point is E,
and once it is done, join them. So, this is the first part, the second part, and this is the third
part; this is the way we trisect an angle.
(Refer Slide Time: 11:26)

If it is something like a circle, the easiest way is if we would like to divide a circle, first
draw a circle. Use your protractor, mark equal angles. So, draw a line, first construct
something like a line, use your protractor, mark particular points easiest way, divide that.
If it is something like we would like to divide this full 360 degrees into 8 equal parts; first,
second, third, fourth, fifth, sixth, seventh, and eighth. The easiest way is 0, 45, 90, and so
on plus addition to 360 if we mark it. Those points we can join it, straight away divide that
circle.

(Refer Slide Time: 12:33)

Now, let us look at how to construct a circle passing through 3 points. For example, we
have point A, we have point B, we have point C, but we don't know how to construct a
circle passing through A, B, C; that is what we are going to learn.
(Refer Slide Time: 13:04)

Now, we know only 3 points A, B, and C; no more details are known. We do not know
even where the centre is, what the radius is, and so on.

(Refer Slide Time: 13:16)

Let us look at the constructional procedure. Points A, B, C are given. To construct the line
first step is join point A and B, to get AB line. This is the line what we know. Let us look
at the constructional procedure. Points A, B, C are given. To construct the line first step is
to join points A and B to get the AB line. This is the line that we know. Similarly, join
points B and C to get line BC; point B and point C, join them. These two lines we got to
know.
Now, pick the first line AB. How we have constructed bisecting AB, use a radius greater
than AB, mark an arc from centre A, with the same radius from B, mark another one.
Similarly, from A centre mark an arc, similarly B with the same radius mark another arc.
So, once we know this point and this point, join them. Perpendicular bisector, we know,
draw it as an extension line. Similarly, construct perpendicular bisector to BC.

(Refer Slide Time: 14:49)

So, this line we have already constructed, to construct BC same procedure. From B greater
than the distance mark an arc, from C as the radius mark an arc; with the same radius mark
from C, mark from B. Wherever these points are in ah joining, construct one more line.
These two perpendicular bisectors, first one and the second perpendicular bisectors, where
they will intersect, will be our centre of the circle. Once centre of the circle is identified,
from centre O, measure the distance AO, whatever that distance, use your compass, which
passes through A, B, C with center O. This is the way we construct it.
(Refer Slide Time: 16:00)

Let us look at that on the sheet. What we know is three points we know; A, perhaps some
point B, maybe some point C. These are the points that we know. Let us join these points,
let us name them as A, B, and C. Join AB, join BC.
Now, draw a perpendicular bisector from centre A, similarly construct from B, identify
points and join these two points. We do not know where exactly centralizing. Similarly,
now from BC, centre B, the intersected points, pick that, join them.
So, it looks like this is the point where they are going to intersect. Let us call that point O.
Now, join AB, that must be the radius. So, for that, you see, there is a circle that is passing
through A, B, and C points with radius O.
If one requires the distance between that AO is the radius of that circle, something like the
leader lines we show, R whatever those units. Let us measure it using our scale. This is
something like 3.4 centimeters. So, R, we use millimeters as the notation; R 34 for that
circle..
(Refer Slide Time: 19:36)

In today's class, we have learned how to bisect an angle, how to trisect an angle, how to
divide a circle, and a circle passing through three points. In the next class, we will learn
more about drawing a normal and a tangent to a circle. Similarly, how to draw a tangent
to a circle from an exterior point and a regular polygon has to be constructed for a given
side, how to do that. These are the things we will learn in lecture 9.
Thank you.