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

Observe the provided figure carefully. A point is a location, a line extends infinitely in both directions, a ray starts from a point and extends infinitely in one direction, and a line segment is a part of a line with two endpoints.

(a) Five points are: O, B, C, D, E.

(b) A line can be named using any two points on it, extending infinitely. For example, the line passing through points O and D can be represented as $\overrightarrow{OD}$.

(c) Four rays are identified by their starting point and another point on the ray. Examples include: $\overrightarrow{OB}$ (starting at O, going through B), $\overrightarrow{OC}$ (starting at O, going through C), $\overrightarrow{OD}$ (starting at O, going through D), and $\overrightarrow{OE}$ (starting at O, going through E).

(d) Five line segments are parts of lines with two endpoints. Examples are: $\overline{DE}$, $\overline{OE}$, $\overline{OC}$, $\overline{OB}$, and $\overline{OD}$.

A line can be named by any two points lying on it. Since there are four points (A, B, C, D) on the line, we can choose any two points to name the line. The possible combinations of two points are:

Using A: AB, AC, AD

Using B: BA, BC, BD

Using C: CA, CB, CD

Using D: DA, DB, DC

In total, there are 12 possible ways to name the line using two letters at a time from the four given points.

By observing the provided figure, we can identify the required geometrical elements:

(a) The line containing point E is the line that passes through E. This line also passes through points A and F. Thus, it can be named as line AE, line FE, or line AF.

(b) The line passing through point A is the same line identified in part (a). It can be named as line AE or line AF.

(c) The line on which point O lies is the line passing through O, B, C, and D. This line can be named as line CO, line OC, line BO, line OB, etc.

(d) Two pairs of intersecting lines are lines that cross each other at a point. From the figure, we can see that line AD intersects line AE at point A. Also, line FE (which is the same as line AE) intersects line CO at point O. Therefore, two pairs of intersecting lines are (AD and AE) and (AE and CO).

(a) An infinite number of lines can pass through a single given point. Imagine a point; you can draw lines through it in every possible direction, extending endlessly.

(b) Only one unique line can pass through two distinct given points. If you have two points, there is only one straight path that connects them directly.

Let's draw the figures based on the given conditions:

(a) Point P lies on line segment AB: Draw a line segment and label its endpoints as A and B. Place a point P somewhere on this segment between A and B.

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

(c) Line $l$ contains E and F but not D: Draw a line and label it as $l$. Mark two points E and F on this line. Mark a point D that is not on line $l$.

(d) Two rays $\overline{OP}$ and $\overline{OQ}$ meet at O: Draw a point O. From O, draw a ray extending in one direction and label a point P on it. From the same point O, draw another ray extending in a different direction and label a point Q on it. Both rays start from O.