Basic Geometrical Ideas
Class 6th Mathematics CBSE Solution
- (a) Five points (b) A line (c) Four rays (d) Five-line segments Use the figure…
- Name the line given in all possible (twelve) ways, choosing only two letters at…
- (a) Line containing point E. (b) Line passing through A. (c) Line on which O…
- How many lines can pass through; (a) one given point? (b) two given points?…
- Draw a rough figure and label suitably in each of the following cases: (a) Point…
- Consider the following figure of line vector mn . Say whether following…
- Classify the following curves as (i) Open or (ii) Closed. g sum _ (d) 1/4)…
- Draw rough diagrams to illustrate the following: (a) Open curve (b) Closed…
- Draw any polygon and shade its interior.
- Consider the given figure and answer the questions: (a) Is it a curve? (b) Is it…
- Illustrate, if possible, each one of the following with a rough diagram: (a) A…
- Name the angles in the given
- In the given diagram, name the point(s) (a) In the interior of DOE (b) In the…
- Draw rough diagrams of two angles such that they have one point in common/three points in…
- Draw a rough sketch of a triangle ABC. Mark a point P in its interior and a…
- Identify three triangles in the figure. b a/d c
- Write the names of seven angles. b a/d c
- Write the names of six-line segments.
- Which two triangles have B as common?
- Draw a rough sketch of a quadrilateral PQRS. Draw its diagonals. Name them. Is…
- Draw a rough sketch of a quadrilateral KLMN. State (a) two pairs of opposite…
- Investigate: Use strips and fasteners to make a triangle and a quadrilateral.…
- From the figure, identify: (a) the centre of the circle (b) three radii (c) a…
- Is every diameter of a circle also a chord?
- Is every chord of a circle also a diameter?
- Draw any circle and mark (a) its centre (b) a radius (c) a diameter (d) a sector…
- Say true or false: (a) Two diameters of a circle will necessarily intersect. (b)…
Exercise 4.1
Question 1.Use the figure to name:
(a) Five points
(b) A line
(c) Four rays
(d) Five-line segments
Answer:
(a) Five points
D, E, O, B and C are the five points in the given figure.
(b) BD is a line in the given figure.
(c) Four rays are;
OE, OD, OC and OB
(d) Five-line segments are;
DE, EO, OB, OC and BE
Question 2.
Name the line given in all possible (twelve) ways, choosing only two letters at a time from the four given.
Answer:
Let’s start with the A,
AB, AC, and AD
BC, BD and BA
CD, CA and CB
DA, DB and DC
So,
AB, AC, AD, BC, BD, BA, CD, CA, CB, DA, DB and DC are the all possible ways to name the given line.
Question 3.
Use the figure to name:
(a) Line containing point E.
(b) Line passing through A.
(c) Line on which O lies.
(d) Two pairs of intersecting lines.
Answer:
(a) Line containing point E is AE.
(b) Line passing through point A is AE.
(c) Line on which O lies is OC.
(d) Two pairs of intersecting lines are;
AE and OC,
AE and EF
Question 4.
How many lines can pass through;
(a) one given point?
(b) two given points?
Answer:
(a) Infinite numbers of lines can pass through a single point.
(b) Only one line can pass through two given points.
Question 5.
Draw a rough figure and label suitably in each of the following cases:
(a) Point P lies on
.
(b) 
and 
intersect at M.
(c) Line l contains E and F but not D.
(d) 
and 
meet at O.
Answer:
(a) Point P liying on AB
(b) XY and PQ intersecting at M
(c) Line containing EF but not D
(d) OP and OQ meeting at O.
Question 6.
Consider the following figure of line 
. Say whether following statements are true or false in context of the given figure:
(a) Q, M, O, N, and P are the points on the line MN.
(b) M, O, N are points on a line segment MN.
(c) M and N are end points of line segment MN.
(d) O and N are end points of line segment OP.
(e) M is one of the end points of line segment QO.
(f) M is point on ray OP.
(g) is different from 
.
(h) is same as 
.
(i) is not opposite to .
(j) O is not an initial point of .
(k) N is the initial point of 
and 
Answer:
(a) True.
As we can see the line is passing through all the points.
(b) True.
As we know that a line segment contains every point on the line between its endpoints. So, MN is a line segment with end points M and N, & has O in between its endpoints only.
(c) True.
As we know that a line segment contains every point on the line between its endpoints and MN is a line segment with end points M and N.
(d) False.
If OP is the line segment than O and N can’t be its endpoints as the Name it self tells us O and P is the endpoints.
(e) False.
M can’t be one of the endpoints of line segment QO as its end points are Q and O itself.
(f) False.
A ray has only one end-point and extends infinitely in the other direction for example sun rays.
So, the ray OP starts from O and extends it self in the direction of P and it doesn’t have M on it.
(g) True.
As Both the rays has different origin points O and Q.
(h) False.
As we can see both the rays are starting from the same point but extending in different directions so they can’t be same.
(i) False.
Ray OM is Opposite to ray OP
(j) False.
O is the Initial point of OP as the name it self suggest it is starting from O only.
(k) True.
N is the initial Point of NP and NM.
Exercise 4.2
Question 1.Classify the following curves as
(i) Open or
(ii) Closed.
Answer:
We can identify it by looking at the figures,
(a) Open
(b) Closed
(c) Open
(d) Closed
(e) Closed
Question 2.
Draw rough diagrams to illustrate the following:
(a) Open curve
(b) Closed curve.
Answer:
(a) Open curve
(b) Closed curve
Question 3.
Draw any polygon and shade its interior.
Answer:
A polygon is a n-sided two-dimensional figure, where n is the number of sides. Here, a pentagon is shown which as 5 sides.
Question 4.
Consider the given figure and answer the questions:
(a) Is it a curve?
(b) Is it closed?
Answer:
(a) Yes, it is a curved figure
(b) Yes, it is closed figure.
Question 5.
Illustrate, if possible, each one of the following with a rough diagram:
(a) A closed curve that is not a polygon.
(b) An open curve made up entirely of line segments.
(c) A polygon with two sides.
Answer:
(a) A closed curve that is not a polygon – a polygon is a two-dimentional figure which has minimum 3 or more straight lines. So, circle isn’t a polygon.
(b) An open curve made up entirely of line segments.
(c) A polygon with two sides isn’t possible. Because a polygon is a closed shape with sides minimum number of three sides.
Exercise 4.3
Question 1.Name the angles in the given
Answer:
∠BAD, ∠ADC, ∠DCB and ∠CBA
Question 2.
In the given diagram, name the point(s)
(a) In the interior of ∠DOE
(b) In the exterior of ∠EOF.
(c) on ∠EOF.
Answer:
(a) Point A is in the interior of ∠DOE
(b) Points C, A and D are in the exterior of ∠EOF.
(c) Point B is on ∠EOF.
Question 3.
Draw rough diagram of two angles such that they have
(a) One point in common
(b) Two points in common
(c) Three points in common
(d) Four points in common
(e) One ray in common.
Answer:
(a) One point in common-
∠POQ and ∠ROS has one point in common i.e., O.
(b) Two points in common
∠AOB and ∠COB has two points in common i.e. O and B.
(c) Three points in common
∠POM and ∠LOM have three points in common i.e. O, M and N.
(d) Four points in common
∠EOB and ∠AOB have four points in common i.e. B, C, D and O.
(e) One ray in common
∠LOM and ∠MON has one common ray i.e. OM.
Exercise 4.4
Question 1.Draw a rough sketch of a triangle ABC. Mark a point P in its interior and a point Q in its exterior. Is the point A in its exterior or in its interior?
Answer:
Point A lies on the given ∆ABC.
Question 2.
Identify three triangles in the figure.
Answer:
∆ABC, ∆ADB and ∆ADC
Question 3.
Write the names of seven angles.
Answer:
∠ABD, ∠ADB, ∠ADC, ∠BDA, ∠BCA, ∠BAC, and ∠CAD
Question 4.
Write the names of six-line segments.
Answer:
AB, AC, AD, BC, BD, and DC
Question 5.
Which two triangles have ∠B as common?
Answer:
∠ABD and ∠ ABC
Exercise 4.5
Question 1.Draw a rough sketch of a quadrilateral PQRS. Draw its diagonals. Name them. Is the meeting point of the diagonals in the interior or exterior of the quadrilateral?
Answer:
PR and QS are two diagonals and they are meeting at points Owhich is the interior point of PQRS.
Question 2.
Draw a rough sketch of a quadrilateral KLMN. State “
(a) two pairs of opposite sides,
(b) two pairs of opposite angles,
(c) two pairs of adjacent sides,
(d) two pairs of adjacent angles.
Answer:
(a) KL and NM,
KN and LM
(b) ∠KLM and ∠KNM
∠NKL and ∠NML
(c) NK, NM and LK, LM
KL, KN and MN, ML
(d) ∠N, ∠K and ∠L , ∠M
∠N, ∠M and ∠K, ∠L
Question 3.
Investigate:
Use strips and fasteners to make a triangle and a quadrilateral.
Try to push inward at any one vertex of the triangle. Do the same to the quadrilateral.
Is the triangle distorted? Is the quadrilateral distorted? Is the triangle rigid?
Why is it that structures like electric towers make use of triangular shapes and not quadrilaterals?
Answer:
This is an interesting activity. The students must try this on their own.
Exercise 4.6
Question 1.From the figure, identify:
(a) the centre of the circle
(b) three radii
(c) a diameter
(d) a chord
(e) two-points in the interior
(f) a point in the exterior
(g) a sector
(h) a segment
Answer:
(a) O
(b) OA, OB and OC
(c) AC
(d) ED
(e) O and P
(f) Q
(g) AOB
(h) DE
Question 2.
Is every diameter of a circle also a chord?
Answer:
Yes, it is true that every diameter of a circle also a chord.
Fact: The diameter is the longest chord in the circle.
Question 3.
Is every chord of a circle also a diameter?
Answer:
No, every chord of a circle is not the diameter.
Because for a chord to be the diameter it should passes through the centre of the circle.
Question 4.
Draw any circle and mark
(a) its centre
(b) a radius
(c) a diameter
(d) a sector
(e) a segment
(f) a point in its interior
(g) a point in its exterior
(h) an arc
Answer:
(a) O
(b) OP
(c) PQ
(d) POR
(e) ST
(f) O
(g) M
(h) PR
Question 5.
Say true or false:
(a) Two diameters of a circle will necessarily intersect.
(b) The centre of a circle is always in its interior.
Answer:
(a) True
Diameters of a circle always intersect each other at the centre of the circle.
(b) True
As it is the centre so it will be in interior only.