ICSE Class 6 Maths Chapter 16 Fundamental Concepts

Unit 4 - Geometry

Chapter 16 - Fundamental Concepts

16.1 Basic Concept

Geometry is the study of the position, shape, size and other properties of different figures. Geometrical terms such as point, line, plane, etc., carry the basic ideas for the development of geometry.

1. Point

A point is a mark of position. It has neither length nor width nor thickness; and occupies no space.

A point is represented by a dot. In general, it is denoted by a capital letter, such as A, P, X, etc. as shown alongside, and is read as 'point P', 'point X' and so on.

2. Line

A line has only length. It has neither width nor thickness.

A line, as shown alongside, is represented by a straight mark with two arrow heads and is denoted by two capital letters.

A line through points A and B is denoted as AB or BA and read as 'line AB' or 'line BA.'

1. The two arrows of a line drawn in opposite directions indicate that the line has unlimited length, i.e. it can be extended up to any distance on either side.

2. A line may be straight or curved, but when we say 'a line', it means a straight line only. The basic idea of a line is its straightness and that it extends infinitely in both directions.

3. An unlimited number of lines can be drawn through a given fixed point. The adjoining figure shows a fixed point A. It is clear from the figure that an unlimited number of lines can be drawn through A.

4. One and only one line can be drawn through any two fixed points. In the adjoining figure, A and B are two fixed points. It is clear from the figure that only one line can be drawn through the fixed points A and B.

5. Every line has an infinite (uncountable) number of points in it.

3. Ray

It is a line, i.e. a straight line, that starts from a given fixed point and moves in the same direction.

The adjoining figure shows a straight line that starts from a given fixed point A and moves through point B in the same direction. Therefore, it is a ray and is written as AB with an arrow on top.

1. A ray has only one end (fixed) point, which is also known as its initial point.

2. A ray extends indefinitely in only one direction.

3. An unlimited number of rays can be drawn with the same initial point. The adjoining figure shows a point source of light .S. SA, SB, SC, SD, etc., are rays of light starting from the source S. Clearly, there are a very large number of such rays each having the same initial point S.

4. A ray is a part of a line.

4. Line Segment

A line segment is a part of a straight line.

The adjoining figure shows a line segment AB, which has two end points A and B. AB and BA both represent the same line segment.

A line segment is a part of a line as well as of a ray.

Make the following facts clear:

(i) Line AB, i.e. AB with arrows on both sides or line BA, i.e. BA with arrows on both sides

(ii) Ray AB = AB with arrow on the right side

(iii) Ray BA = BA with arrow on the left side

(iv) Line segment AB (or BA)

Clearly, line segment AB (or BA) is a part of both the rays AB (AB with arrow) and BA (BA with arrow).

At the same time, the line segment AB (or BA) is a part of both the lines AB (AB with arrows) and BA (BA with arrows).

5. Surface

A surface has length and width, but no thickness.

A page of your book, the outside of a box, a black-board, etc., represent surfaces.

A surface may be flat or curved.

The surface of a wall is flat, while the surface of a cricket ball is curved.

6. Plane

It is a flat surface. A plane has length and width, but no thickness.

A page of your book, the surface of a wall, the top of a table, etc., are some examples of planes.

(i) When a straight line is drawn through any two points on a plane, the line lies entirely within in the plane.

(ii) A plane is a surface that extends indefinitely in all directions.

7. Parallel Lines

Two straight lines are said to be parallel to each other if they lie in the same plane and do not meet even when produced upto any extent on either side.

The given figure shows two lines AB and CD that are parallel to each other.

The line AB is parallel to the line CD is symbolically represented by writing AB // CD, read as line AB is parallel to line CD.

8. Intersecting Lines

If two lines lie in the same plane and are not parallel to each other, they are called intersecting lines.

The figure given above shows two lines AB and CD that are not parallel. Such lines either intersect at point O or will intersect at point O if extended.

1. The distance between two parallel lines is always the same whereas the distance between two non-parallel lines (lines that are not parallel) keeps changing.

2. Two different lines in a plane are either parallel or they intersect at only one point.

3. The definitions of parallel lines and intersecting lines are applicable only when the lines are in the same plane, i.e. when the lines are co-planar.

9. Collinear Points

If three or more points lie on the same straight line, then the points are called collinear points.

The given figure shows the collinear points A, B, C and D; as these points lie on the same line.

The adjoining figure shows three non-collinear points: P, Q and R. These points do not lie on the same straight line.

10. Concurrent Lines

If three or more straight lines (in the same plane) pass through the same point, the lines are called concurrent lines and the point is called the point of concurrence.

The given figure shows the concurrent lines AB, CD, EF and GH; all these lines are in the same plane and pass through the same point P. Clearly, point P is the point of concurrence.

Exercise 16(A)

1. State true or false, if false, correct the statement:

(i) A dot has width but no length.

(ii) A ray has an infinite length only on one side of it.

(iii) A line segment PQ is written as PQ with an arrow on top.

(iv) PQ with arrows on both sides represents a straight line

(v) Three points are said to be collinear if they lie in the same plane

(vi) Three or more points, all lying in the same line, are called collinear points.

2. Write how many lines can be drawn through:

(i) a given point?

(ii) two given fixed points?

(iii) three collinear points?

(iv) three non-collinear points?

3. The shaded region of the given figure shows a plane:

(a) Name:

(i) three collinear points.

(ii) three non-collinear points.

(iii) a pair of intersecting lines.

(b) State whether true or false:

(i) Line DE is contained in the given plane p.

(ii) Lines AB and DE intersect at point C.

(iii) Points D, B and C are collinear.

(iv) Points D, B and E are collinear.

4. Correct the statement if it is wrong:

(i) A ray can be extended infinitely on either side.

(ii) A ray has a definite length.

(iii) A line segment has a definite length.

(iv) A line has two end points.

(v) A ray has only one end point.

5. State true or false, if false, give the correct statement:

(i) A line has a countable number of points in it.

(ii) Only one line can pass through a given point.

(iii) The intersection of two planes is a straight line.

6. State whether the following pairs of lines or rays appear to be parallel or intersecting.

If, on extending the two lines on either side, they intersect or appear to intersect, they are intersecting lines.

7. Give two examples, from your surroundings, for each of the following:

(i) points

(ii) line segments

(iii) plane surfaces

(iv) curved surfaces.

8. Under what condition will two straight lines in the same plane have:

(i) no point in common.

(ii) only one point in common.

(iii) an infinite number of points in common.

If possible, draw diagrams in support of your answer.

9. Mark two points A and B on a page of your exercise book. Mark a third point P such that:

(i) P lies between A and B and the three points A, P and B are collinear.

(ii) P does not lie between A and B yet the three points are collinear.

(iii) the three points do not lie in a line.

10. Mark two points P and Q on a piece of paper. How many lines can you draw:

(i) passing through both the points P and Q?

(ii) passing through the point P?

(iii) passing through the point Q?

11. The adjoining figure shows a line AB. Draw figures to represent:

(i) ray AB, i.e. AB with arrow on top

(ii) BA with arrow on top

(iii) line segment AB.

12. The adjoining figure shows a ray AB. Draw figures to show:

(i) ray BA, i.e. BA with arrow on top

(ii) line AB

(iii) line segment BA.

13. The adjoining figure shows a line segment AB. Draw figures to represent:

(i) ray AB, i.e. AB with arrow on top

(ii) line AB, i.e. AB with arrows on both sides

(iii) ray BA.

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