Rd Sharma 2020 2021 for Class 6 Maths Chapter 10 - Basic Geometrical Concepts

Answer:

Answer:

If we join the points AM, HE, TO, RUN, IF,  6 line segments can be formed.

These six line segments are AM, HE, TO, RU, UN and IF.

Answer:

(i) Five line segments are PQ, RS, PR, QS and AP.

(ii) Five rays are $\underset{\mathrm{QC},}{\to }\underset{\mathrm{SD},}{\to }\underset{\mathrm{PA},}{\to }\underset{\mathrm{RB}}{\to }$ and $\underset{\mathrm{RA}}{\to }$.

(iii) Non-intersecting line segments are PR and QS.

Answer:

(i) Yes, we can draw line segments on the face of a cuboid.

(ii) No, we cannot draw a line segment on the surface of an egg or apple.

(iii) Yes, we can draw line segments on the curved surface of a cylinder.
      Every line segment parallel to the axis of a cylinder on the curved surface will be a line segment.

(iv) Yes, we can draw line segments on the curved surface of a cone.
      Every line segment joining the vertex of a cone and any point on the circumference of the cone will be a line segment.

(v) Yes, we can draw line segments on the base of a cone.

Answer:

If there are n points in a plane and no three of them are collinear, the number of line segments obtained by joining these points is equal to $\frac{n(n-1)}{2}$.
On applying the above formula, we get:

(i) For two points A and B:
     Number of line segments = $\frac{2(2-1)}{2}=1$

(ii) For three non-collinear points A, B and C:

      Number of line segments = $\frac{3(3-1)}{2}=\frac{3\times 2}{2}=3$

(iii) For four points such that no three of them belong to the same line:

       Number of line segments = $\frac{4(4-1)}{2}=\frac{4\times 3}{2}=6$

(iv) For any five points so that no three of them are collinear:

       Number of line segments = $\frac{5(5-1)}{2}=\frac{5\times 4}{2}=10$

Answer:

Line segments in the given figure are AB, AC, AD, AE, BC, BD, BE, CD, CE and DE.
Thus, there are 10 line segments.

Answer:

Name of all rays with initial point as A :  $\underset{\mathrm{AP}}{\to }$  and  $\underset{AB}{\to } or \underset{AC}{\to } or \underset{AQ}{\to }$
Name of all rays with initial point as B : $\underset{\mathrm{BP}, }{\to } or \underset{\mathrm{BA},}{\to }$ and $\underset{BC }{\to } or \underset{BQ}{\to }$
Name of all rays with initial point as C : $\underset{\mathrm{CP},}{\to } or \underset{CA}{\to } or \underset{CB}{\to }$ and $\underset{\mathrm{CQ}}{\to }$

(i) No, because the origin point of both the rays $\underset{AB}{\to }$ and $\underset{AC}{\to }$ is same.
(ii) Yes, because the origin point of both the rays $\underset{BA}{\to }$ and $\underset{CA}{\to }$ is different.
(iii) Yes, because both the rays $\underset{CP}{\to }$ and $\underset{CQ}{\to }$ are in opposite directions.

Answer:

Examples of line segments in our home is:
(i) grout lines in the tile floors
(ii) groves where wooden flooring connects
(iii) metal outline of a sliding glass door

Answer:

(i) Open curve
    
(ii) Closed curve
     

Answer:

(i) Open
(ii) Closed
(iii) Closed
(iv) Open
(v) Open
(vi) Closed

Answer:

ABCD is a polygon and AC and BD are its two diagonals.

Answer:

(i) A circle is a simple closed curve but not a polygon. A polygon has line-segments, but a circle only has a curve.
    
(ii) Rough diagram of an open curve made up entirely of line segments:
     
     
(iii) A polygon with two sides is not possible. 

Answer:

Three points, namely A, P and H can be marked as follows:

Answer:

Let us draw a line and name it l.

Answer:

Let us first draw a line. Two points on it are P and Q. Now, the line can be written as line PQ.

Answer:

(i) Three examples of points are:
The period at the end of a sentence, a pinhole on a map and the point at which two walls and the floor meet at the corner of the room

(ii) Three examples of portion of a line are:
Tightly stretched power cables, laser beams and thin curtain rods

(iii) Three examples of plane of a surface are:
The surface of a smooth wall, the surface of the top of a table and the surface of a smooth white board

(iv) Three examples of portion of a plane are:
The surface of a sheet of a paper, the surface of a calm water in a swimming pool and the surface of a mirror

(v) Three examples of curved surfaces are:
The surface of a gas cylinder, the surface of a tea pot and the surface of an ink pot

Answer:

(i) Yes
(ii) No
(iii) Yes

Answer:

No, we cannot draw a line on the surface of a sphere, which lies wholly on it.

Answer:

Unlimited number of lines can be drawn passing through a point L.

Answer:

We have two points P and Q and we draw a line passing through these two points.

Only one line can be drawn passing through these two points.

Answer:

Ceiling of a room is an example of a horizontal plane in our environment.
Wall of a room is an example of a vertical plane in our environment.

Answer:

Lines passing through one point: Unlimited


Lines passing through two points: One


Lines passing through any three collinear points : One

Answer:

Yes, it is possible if the three points lie on a straight line.

Answer:

The length of a line is infinite. Thus, it is not possible to find its mid-point. 
On the other hand, we can find the mid-point of a line segment.

Answer:

There  are three non-collinear points A, B and C.
Three lines can be drawn through these three points. These three lines are AB, BC and AC.

Answer:

(i) Yes
     A group of points that lie in the same plane are called coplanar points.
     Thus, it is possible that 150 points can be coplanar.

(ii) No
      3 points will be coplanar because we can have a plane that can contain all 3 points on it.
      Thus, it is not possible that 3 points will be non-coplanar.
     

Answer:

From the given figure, we can observe that:

(i) D, A and C are collinear points.
(ii) A, B and C are non-collinear points.
(iii) A, B and E are collinear points.
(iv) B, C and E are non-collinear points.

Answer:

No p, q and r are not necessarily coplanar .
e.g.:-If we take p as intersecting line of two consecutive walls of a room , q as a line on the first wall and r on the second wall whose(both walls) intersection is line p .
Then we can see that p , q and r are not coplanar.

Answer:

(i) Three examples of intersecting lines in our environment:
    

(ii) Three examples of parallel lines in our environment:
     

Answer:

We have:

(i) All pairs of parallel lines: (l, m), (m, n) and  (l, n)
(ii) All pairs of intersecting lines: (l, p), (m, p), (n, p), (l, r), (m, r), (n, r), (l, q), (m, q), (n, q), (q, p) and (q, r)
(iii) Lines whose point of intersection is I: (m, p)
(iv) Lines whose point of intersection is D: (l, r)
(v) Lines whose point of intersection is E: (m, r)
(vi) Lines whose point of intersection is A: (l, q)
(vii) Collinear points: (G, A, B and C), (D, E, J and F), (G, H, I , J and K), (A, H and D), (B, I and E), and (C, F and K)

Answer:

From the given figure, we have:

Concurrent lines can be defined as three or more lines which share the same meeting point.
Clearly lines n, q and l are concurrent with A as the point of concurrence.
Lines m, q and p are concurrent with B as the point of concurrence.

Answer:

From the given figure, we have:

(i) Six lines can be drawn through these four points as given in the figure.
(ii) These lines are AB, BC, CD, AD, BD and AC.
(iii) Lines which are concurrent at A are AC, AB and AD.

Answer:

Maximum number of points of intersection of three lines in a plane will be three.


Minimum number of points of intersection of three lines in a plane will be zero.

Answer:

Maximum number of points of intersection of four lines in a plane will be six.



Minimum number of points of intersection of four lines in a plane will be zero.

Answer:

We have:

Thus, lines p, q and r intersect at a common point O.
Also, lines p, r and s are concurrent.
Therefore, lines p, r and s intersect at a common point. But  q and r intersect each other at O.
So, s, q and r intersect at O.
Hence, p, q, r and s are concurrent. Lines p, q, r and s intersect at O.

Answer:

Lines p, q, and r are concurrent. So, lines p, q and r intersect at a common point O.

Given lines p, s and t are concurrent. So, lines p, s and t also intersect at a common point.
However, it is not always true that q, r and s or q, r and t are concurrent.

Answer:

(i) A page of a book is a physical example of a plane.
(ii) An ink pot has both curved and plane surfaces.
(iii) Two lines in a plane are either parallel or are intersecting.

Answer:

(i) False

(ii) True

(iii) False
They may be parallel.

(iv) True

(v) False
In every vertical plane there must be horizontal line.
e.g.:-Intersecting line of a wall and floor of a room is horizontal line and it is also a line on the wall that is on the vertical plane.

(vi) False
Counter example:-Intersecting line of a wall and floor of a room is horizontal line and it is also a line on the wall that is on the vertical plane.

(vii) True

(viii) False
They can be parallel line also.

(ix) False
Through a point of concurrence, at least three lines should pass.

(x) True

(xi) False
Here ABCD is in one plane and DPQ may be in different plane and they may be intersecting through the point D.


(xii) False
Two lines are either intersecting at one point or they may be parallel lines or they may be coincident lines.

(xiii) False
Through a given point infinite line can be drawn.

(xiv) False
Points are collinear only if all points are in the same plane.

(xv) True

(xvi) False
Minimum number of points of intersection of three lines be zero.