CBSE Class 6 Mathematics Chapter 4: Basic Geometrical Ideas NCERT Solutions

Observing the provided figure, we can identify the geometric elements as follows:

(a) Five points: Points are specific locations. From the figure, we can name five distinct points as O, B, C, D, and E.

(b) A line: A line is a straight path that extends infinitely in both directions. It can be named using any two points on the line. For example, the line passing through points O and D can be represented as $\backslash overleftrightarrow\{OD\}$.

(c) Four rays: A ray has one endpoint and extends infinitely in one direction. We can identify rays starting from a point and going in a specific direction. For instance, starting from point O, we have rays $\backslash overrightarrow\{OB\}$, $\backslash overrightarrow\{OC\}$, $\backslash overrightarrow\{OD\}$, and $\backslash overrightarrow\{OE\}$.

(d) Five line segments: A line segment is a part of a line that has two endpoints. We can name several line segments by choosing pairs of points. Examples include $\backslash overline\{OB\}$, $\backslash overline\{OC\}$, $\backslash overline\{OD\}$, $\backslash overline\{OE\}$, and $\backslash overline\{DE\}$.

We are given four points A, B, C, and D that lie on the same line. To name the line using two letters at a time, we consider all possible pairs of these points. The order of the letters does not matter when naming a line segment, but it does when naming a line or ray. However, the question asks for naming the line in all possible ways using two letters, implying combinations of points that define the line.

The possible pairs of points are:

AB, AC, AD, BC, BD, CD.

Since these points are collinear, each pair defines the same line. If we consider the order of points to name the line, we can also have:

BA, CA, DA, CB, DB, DC.

Therefore, the line can be named in twelve possible ways by choosing two letters at a time: AB, AC, AD, BC, BD, CD, BA, CA, DA, CB, DB, DC.

Based on the provided figure:

(a) Line containing point E: The line that passes through point E also passes through points A and F. Thus, the line containing point E can be named as line AE or line FE, or $\backslash overleftrightarrow\{AE\}$ or $\backslash overleftrightarrow\{FE\}$.

(b) Line passing through A: The line that passes through point A also passes through points E and F. This line can be named as line AE or line AF, or $\backslash overleftrightarrow\{AE\}$ or $\backslash overleftrightarrow\{AF\}$.

(c) Line on which O lies: Point O lies on the line segment CO. This line can be named as line CO or line OC, or $\backslash overleftrightarrow\{CO\}$ or $\backslash overleftrightarrow\{OC\}$.

(d) Two pairs of intersecting lines: Intersecting lines are lines that cross each other at a single point. From the figure, we can see that line AD intersects line FE at point E. Also, line CO intersects line AE (or AF) at point O. Therefore, two pairs of intersecting lines are (AD and FE) and (CO and AE).

(a) One given point: An infinite number of lines can be drawn passing through a single given point. Imagine a point on a paper; you can draw lines through it in every possible direction.

(b) Two given points: Exactly one unique line can be drawn passing through two distinct given points. If you have two points, there is only one straight path that connects them and extends infinitely in both directions.

Let's draw and label the figures as per the given conditions:

(a) Point P lies on line segment AB: We draw a line segment and label its endpoints as A and B. Then, we place a point P somewhere on this segment between A and B.

(b) Line segments $\backslash overline\{XY\}$ and $\backslash overline\{PQ\}$ intersect at M: We draw two line segments, one named XY and the other named PQ. We ensure they cross each other at a point, and we label this intersection point as M.

(c) Line l contains E and F but not D: We draw a line and label it as l. We place points E and F on this line. We also place a point D, ensuring it is not on line l.

(d) Line segments $\backslash overline\{OP\}$ and $\backslash overline\{OQ\}$ meet at O: We draw two line segments that share a common endpoint, O. One segment is OP, and the other is OQ. They meet at their common starting point, O.