Solution:<\/strong><\/p>\n\n\n\n The three points A, P and H are marked as given below:<\/p>\n\n\n\n 2. Draw a line in your notebook and name it using a small letter of the alphabet.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n The line AB is drawn and is named as l<\/p>\n\n\n\n 3. Draw a line in your notebook and name it by taking any two points on it.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Construct a line and name the points PQ. It can be called as the line PQ.<\/p>\n\n\n\n 4. Give three examples from your environment of:<\/strong><\/p>\n\n\n\n (i) Points<\/strong><\/p>\n\n\n\n (ii) Portion of a line<\/strong><\/p>\n\n\n\n (iii) Plane surfaces<\/strong><\/p>\n\n\n\n (iv) Portion of a plane<\/strong><\/p>\n\n\n\n (v) Curved surfaces<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n (i) Points<\/p>\n\n\n\n The three examples are<\/p>\n\n\n\n Pinhole on the map<\/p>\n\n\n\n Two walls and floor meeting at the corner<\/p>\n\n\n\n Period at the end of the sentence<\/p>\n\n\n\n (ii) Portion of a line<\/p>\n\n\n\n The three examples are<\/p>\n\n\n\n Thin curtain rods<\/p>\n\n\n\n Laser beams<\/p>\n\n\n\n Stretched power cables<\/p>\n\n\n\n (iii) Plane surfaces<\/p>\n\n\n\n The three examples are<\/p>\n\n\n\n Surface of a white board<\/p>\n\n\n\n Top of a table<\/p>\n\n\n\n Surface of a wall<\/p>\n\n\n\n (iv) Portion of a plane<\/p>\n\n\n\n The three examples are<\/p>\n\n\n\n Surface of a mirror<\/p>\n\n\n\n Calm water in a swimming pool<\/p>\n\n\n\n Surface of the sheet of paper<\/p>\n\n\n\n (v) Curved surfaces<\/p>\n\n\n\n The three examples are<\/p>\n\n\n\n Ink pot<\/p>\n\n\n\n Tea pot<\/p>\n\n\n\n Gas cylinder<\/p>\n\n\n\n 5. There are a number of ways by which we can visualise a portion of a line. State whether the following represent a portion of a line or not:<\/strong><\/p>\n\n\n\n (i) A piece of elastic stretched to the breaking point.<\/strong><\/p>\n\n\n\n (ii) Wire between two electric poles.<\/strong><\/p>\n\n\n\n (iii) The line thread by which a spider lowers itself.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n (i) Yes.<\/p>\n\n\n\n (ii) No.<\/p>\n\n\n\n (iii) Yes.<\/p>\n\n\n\n 6. Can you draw a line on the surface of a sphere which lies wholly on it?<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n No. A line cannot be drawn on the surface of the sphere which lies wholly on it.<\/p>\n\n\n\n 7. Mark a point on a sheet of paper and draw a line passing through it. How many lines can you draw through this point?<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Unlimited number of lines can be drawn through the point L.<\/p>\n\n\n\n 8. Mark any two points P and Q in your note book and draw a line passing through the points. How many lines can you draw passing through both the points?<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Draw a line passing through the points P and Q<\/p>\n\n\n\n Only one line can be drawn passing through this both points.<\/p>\n\n\n\n 9. Give an example of a horizontal plane and a vertical plane from your environment.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n The example of horizontal plane is ceiling of a room.<\/p>\n\n\n\n The example of a vertical plane is wall of a room.<\/p>\n\n\n\n 10. How many lines may pass through one given point, two given points, any three collinear points?<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n The lines passing through one given point is unlimited.<\/p>\n\n\n\n\n\n The lines passing through two given points are only one.<\/p>\n\n\n\n The lines passing through three collinear points are one.<\/p>\n\n\n\n 11. Is it ever possible for exactly one line to pass through three points?<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Yes. It is possible for a line to pass through three points, if the points lie on a straight line.<\/p>\n\n\n\n 12. Explain why it is not possible for a line to have a mid-point.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n We know that the length of the line is infinite and it is not possible to find the midpoint.<\/p>\n\n\n\n But it is possible to find the midpoint of line segments.<\/p>\n\n\n\n 13. Mark three non-collinear points A, B and C in your note book. Draw lines through these points taking two at a time. Name these lines. How many such different lines can be drawn?<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n It is given that the three collinear points are A, B and C<\/p>\n\n\n\n We know that three lines namely AB, BC and AC can be drawn using these points.<\/p>\n\n\n\n 14. Coplanar points are the points that are in the same plane. Thus,<\/strong><\/p>\n\n\n\n (i) Can 150 points be coplanar?<\/strong><\/p>\n\n\n\n (ii) Can 3 points be non-coplanar?<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n (i) Yes. We know that the group of points which lie in the same plane are coplanar points.<\/p>\n\n\n\n Hence, 150 points can be coplanar.<\/p>\n\n\n\n (ii) No. We know that 3 points can be coplanar as we can have plane which contains 3 points.<\/p>\n\n\n\n Hence, 3 points cannot be non-coplanar.<\/p>\n\n\n\n 15. Using a ruler, check whether the following points given in Fig. 10.30 are collinear or not:<\/strong><\/p>\n\n\n\n (i) D, A and C<\/strong><\/p>\n\n\n\n (ii) A, B and C<\/strong><\/p>\n\n\n\n (iii) A, B and E<\/strong><\/p>\n\n\n\n (iv) B, C and E<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n (i) The points D, A and C are collinear.<\/p>\n\n\n\n (ii) The points A, B and C are non-collinear.<\/p>\n\n\n\n (iii) The points A, B and E are collinear.<\/p>\n\n\n\n (iv) The points B, C and E are non-collinear.<\/p>\n\n\n\n 16. Lines p, q are coplanar. So are the lines p, r. Can we conclude that the lines p, q, r are coplanar?<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Yes. The lines p, q and r are coplanar.<\/p>\n\n\n\n 17. Give three examples each of:<\/strong><\/p>\n\n\n\n (i) intersecting lines<\/strong><\/p>\n\n\n\n (ii) parallel lines from your environment.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n (i) Three examples of intersecting lines are<\/p>\n\n\n\n (ii) Three examples of parallel lines from your environment are<\/p>\n\n\n\n 18. From Fig. 10.21, write<\/strong><\/p>\n\n\n\n (i) all pairs of parallel lines.<\/strong><\/p>\n\n\n\n (ii) all pairs of intersecting lines.<\/strong><\/p>\n\n\n\n (iii) lines whose point of intersection is I.<\/strong><\/p>\n\n\n\n (iv) lines whose point of intersection is D.<\/strong><\/p>\n\n\n\n (v) lines whose point of intersection is E.<\/strong><\/p>\n\n\n\n (vi) lines whose point of intersection is A.<\/strong><\/p>\n\n\n\n (vii) collinear points.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n (i) All pairs of parallel lines are (l, m); (m, n); ( l, n).<\/p>\n\n\n\n (ii) All pairs of intersecting lines are (l, p); (m, p); (n, p); ( l, r); (m, r); (n, r); (p, r); (l, q); (m, q); (n, q); (q, p); (q, r).<\/p>\n\n\n\n (iii) Lines whose point of intersection is I are m, p.<\/p>\n\n\n\n (iv) Lines whose point of intersection is D are l, r.<\/p>\n\n\n\n (v) Lines whose point of intersection is E are m, r.<\/p>\n\n\n\n (vi) Lines whose point of intersection is A are l, q.<\/p>\n\n\n\n (vii) Collinear points are (G, A, B, C); (D, E, J, F); (G, H, I, J, K); (A, H, D); (B, I, E); (C, F, K).<\/p>\n\n\n\n 19. From Fig. 10.22, write concurrent lines and their points of concurrence.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n We know that the three or more lines which have the same meeting point are concurrent lines.<\/p>\n\n\n\n Here, the lines n, q and l are concurrent with A as the point of concurrence.<\/p>\n\n\n\n We know that m, q and p are concurrent with B as the point of concurrence.<\/p>\n\n\n\n 20. Mark four points A, B, C and D in your notebook such that no three of them are collinear. Draw all the lines which join them in pairs as shown in Fig. 10.23.<\/strong><\/p>\n\n\n\n (i) How many such lines can be drawn?<\/strong><\/p>\n\n\n\n (ii) Write the names of these lines.<\/strong><\/p>\n\n\n\n (iii) Name the lines which are concurrent at A.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n (i) Six lines can be drawn through the points A, B, C and D.<\/p>\n\n\n\n (ii) The names of these lines are AB, BC, CD, AD, BD and AC.<\/p>\n\n\n\n (iii) AC, AB and AD are the lines which are concurrent at A.<\/p>\n\n\n\n 21. What is the maximum number of points of intersection of three lines in a plane? What is the minimum number?<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n The maximum number of points of intersection of three lines in a plane is three.<\/p>\n\n\n\n The minimum number of points of intersection of three lines in a plane is zero.<\/p>\n\n\n\n 22. With the help of a figure, find the maximum and minimum number of points of intersection of four lines in a plane.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n The maximum number of points of intersection of four lines in a plane is six.<\/p>\n\n\n\n The minimum number of points of intersection of four lines in a plane is zero.<\/p>\n\n\n\n 23. Lines p, q and r are concurrent. Also, lines p, r and s are concurrent. Draw a figure and state whether lines p, q, r and s are concurrent or not.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n We know that the lines p, q and r intersect at the point O.<\/p>\n\n\n\n So the lines p, r and s are concurrent.<\/p>\n\n\n\n The lines p, r, s and p, q, r intersect at O.<\/p>\n\n\n\n Therefore, the lines p, q, r and s are concurrent intersecting at the point O.<\/p>\n\n\n\n 24. Lines p, q and r are concurrent. Also lines p, s and t are concurrent. Is it always true that the lines q, r and s will be concurrent? Is it always true for lines q, r and t?<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n We know that the lines p, q and r intersect at the point O and are concurrent.<\/p>\n\n\n\n The lines p, s and t intersect at the common point and are concurrent. We know that it is not always true that the lines q, r and s or q, r and t are concurrent.<\/p>\n\n\n\n 25. Fill in the blanks in the following statements using suitable words:<\/strong><\/p>\n\n\n\n (i) A page of a book is a physical example of a \u2026.<\/strong><\/p>\n\n\n\n (ii) An inkpot has both \u2026\u2026. Surfaces<\/strong><\/p>\n\n\n\n (iii) Two lines in a plane are either \u2026\u2026. or are \u2026.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n (i) A page of a book is a physical example of a plane.<\/p>\n\n\n\n (ii) An inkpot has both curved and plane surfaces.<\/p>\n\n\n\n (iii) Two lines in a plane are either parallel or are intersecting.<\/p>\n\n\n\n 26. State which of the following statements are true (T) and which are false (F):<\/strong><\/p>\n\n\n\n (i) Point has a size because we can see it as a thick dot on paper.<\/strong><\/p>\n\n\n\n (ii) By lines in geometry, we mean only straight lines.<\/strong><\/p>\n\n\n\n (iii) Two lines in a plane always intersect in a point.<\/strong><\/p>\n\n\n\n (iv) Any plane through a vertical line is vertical.<\/strong><\/p>\n\n\n\n (v) Any plane through a horizontal line is horizontal.<\/strong><\/p>\n\n\n\n (vi) There cannot be a horizontal line in a vertical plane.<\/strong><\/p>\n\n\n\n (vii) All lines in a horizontal plane are horizontal.<\/strong><\/p>\n\n\n\n (viii) Two lines in a plane always intersect in a point.<\/strong><\/p>\n\n\n\n (ix) If two lines intersect at a point P, then P is called the point of concurrence of the two lines.<\/strong><\/p>\n\n\n\n (x) If two lines intersect at a point P, then P is called the point of intersection of the two lines.<\/strong><\/p>\n\n\n\n (xi) If A, B, C and D are collinear points D, P and Q are collinear, then points A, B, C, D, P and Q are always collinear.<\/strong><\/p>\n\n\n\n (xii) Two different lines can be drawn passing through two given points.<\/strong><\/p>\n\n\n\n (xiii) Through a given point only one line can be drawn.<\/strong><\/p>\n\n\n\n (xiv) Four points are collinear if any three of them lie on the same line.<\/strong><\/p>\n\n\n\n (xv) The maximum number of points of intersection of three lines is three.<\/strong><\/p>\n\n\n\n (xvi) The maximum number of points of intersection of three lines is one.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n (i) False<\/p>\n\n\n\n (ii) True<\/p>\n\n\n\n (iii) False<\/p>\n\n\n\n (iv) True<\/p>\n\n\n\n (v) True<\/p>\n\n\n\n (vi) False<\/p>\n\n\n\n (vii) False<\/p>\n\n\n\n (viii) False<\/p>\n\n\n\n (ix) False<\/p>\n\n\n\n (x) True<\/p>\n\n\n\n (xi) False<\/p>\n\n\n\n (xii) True<\/p>\n\n\n\n (xiii) False<\/p>\n\n\n\n (xiv) False<\/p>\n\n\n\n (xv) True<\/p>\n\n\n\n (xvi) False<\/p>\n\n\n\n 27. Give the correct matching of the statements of Column A and Column B.<\/strong><\/p>\n\n\n\n Column A Column B<\/strong><\/p>\n\n\n\n (i) Points are collinear (a) may be parallel or intersecting<\/strong><\/p>\n\n\n\n (ii) Line is completely known (b) are undefined terms in geometry<\/strong><\/p>\n\n\n\n (iii) Two lines in a plan (c) if they lie on the same line<\/strong><\/p>\n\n\n\n (iv) Relations between points and lines (d) can pass through a point<\/strong><\/p>\n\n\n\n (v) Three non-collinear points (e) determine a plane<\/strong><\/p>\n\n\n\n (vi) A plane extends (f) are called incidence properties<\/strong><\/p>\n\n\n\n (vii) Indefinite number of lines (g) if two points are given<\/strong><\/p>\n\n\n\n![\"\"](\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2021\/10\/rd-sharma-solutions-class-6-chapter-10-ex-10-1-image-1.png\" "\\"RD")
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