RS Aggarwal Solutions for Class 6 Chapter 10 Ratio, Proportion and Unitary Method

Exercise 10.1

Question 1. Make three points in your notebook and name them.
Answer: You can mark three points, such as A, P, and H, in your notebook as shown below:

A •
P • • H
In simple words: A point is a small dot that shows an exact location. You can mark points anywhere on paper and give them names using capital letters.

Exam Tip: Always use capital letters (A, B, C, etc.) to name points, and mark them clearly with a small dot so they are easy to see.

Question 2. Draw a line in your notebook and name it using a small letter of the alphabet
Answer: You can draw a line and call it l as shown below:

A ←———— B ———→ l

In simple words: A line extends endlessly in both directions and has no starting or ending point. You can name a line using a lowercase letter like l, m, or n.

Exam Tip: Remember that a line goes on forever in both directions - it is different from a line segment which has two endpoints.

Question 3. Draw a line in your notebook and name it by using two points on it
Answer: You can draw a line first. Two points on it are P and Q. The line can be written as line PQ.
In simple words: When a line passes through two known points, you can name that line using those two point names. Put the two letters together, like PQ or AB.

Exam Tip: When naming a line using two points, make sure both points actually lie on that line, and the order doesn't matter (PQ is the same as QP).

Question 4. Give three examples from your environment of:
(i) Points
(ii) Portion of a line
(iii) Plane of a surface
(iv) Portion of a plane
(v) Curved surface
Answer:
(i) Points - The dot at the end of a sentence, a pinhole on a map, and the corner where two walls and the floor meet in a room.
(ii) Portion of a line - Tightly stretched power cables, laser beams, and thin curtain rods.
(iii) Plane of a surface - The surface of a smooth wall, the surface of the top of a table, and the surface of a smooth white board.
(iv) Portion of a plane - The surface of a sheet of paper, the surface of still water in a swimming pool, and the surface of a mirror.
(v) Curved surface - The surface of a gas cylinder, the surface of a tea pot, and the surface of an ink pot.
In simple words: Points are tiny dots showing exact spots. Line portions are like stretched strings. Flat planes are smooth surfaces that extend endlessly. Curved surfaces bend and are not flat anywhere.

Exam Tip: Choose real-world examples that everyone can visualize - use objects found in school, home, or nature to make your answer clear and relatable.

Question 5. There are a number of ways by which we can visualize a portion of a line. State whether the following represent a portion of line or not:
(i) A piece of elastic stretched to the breaking point
(ii) Wire between two electric poles
(iii) The line thread by which a spider lowers itself
Answer:
(i) Yes
(ii) No
(iii) Yes
In simple words: Elastic and spider silk stretch in straight lines like line segments. But wire sags and curves slightly because of its own weight, so it doesn't represent a straight line portion.

Exam Tip: A true line portion must be perfectly straight with no curves or sags - think of it as having constant tension.

Question 6. Can you draw a line on the surface of a sphere which lies wholly on it?
Answer: No, you cannot draw a line on the surface of a sphere that lies wholly on it because a sphere has a curved surface.
In simple words: A sphere is round in all directions. A straight line only exists on flat surfaces. On a curved ball, any line you draw will curve with the surface.

Exam Tip: Remember that straight lines only lie flat on plane surfaces - curved surfaces cannot contain any straight lines fully on them.

Question 7. Make appoint on the sheet of a paper and draw a line passing through it. How many lines can you draw through this point?
Answer: An unlimited number of lines can be drawn passing through a single point L.

The diagram shows multiple lines (p, n, m, q) all passing through point L in different directions.
In simple words: When you have just one point, you can draw as many different lines as you want through it - there is no limit. Lines can go in any direction from that point.

Exam Tip: This is a key concept - infinitely many lines pass through any single point, but only one line passes through two distinct points.

Question 8. Mark any two points P and Q in your notebook and draw a line passing through the points. How many lines can you draw passing through this both points?
Answer: We have two points P and Q, and we draw a line passing through these two points. Only one line can be drawn passing through these two points.
In simple words: Once you fix two different points, there is only one straight line that can pass through both of them. No other line can go through those exact two spots.

Exam Tip: This is a fundamental rule in geometry - two distinct points always define a unique straight line.

Question 9. Give an example of the horizontal plane and a vertical plane from your environment.
Answer: The ceiling of a room is an example of a horizontal plane in our environment. The wall of a room is an example of a vertical plane in our environment.
In simple words: A horizontal plane runs left and right, like the floor or ceiling. A vertical plane stands up straight, like a wall or door.

Exam Tip: Horizontal means level like water, vertical means standing upright - use everyday objects to identify both types.

Question 10. How many lines may pass through one given point, two given point, any three collinear points?
Answer: Lines passing through one point - unlimited.

Lines passing through two points - one.

Lines passing through any three collinear points - one.
In simple words: A single point lets you draw endless lines. Two points fix exactly one line. Three or more points on the same line still form just one line.

Exam Tip: The key difference is that additional points don't create new lines if they already lie on the same straight path.

Question 11. Is it ever possible for exactly one line to pass through three points?
Answer: Yes, it is possible if three points lie on a straight line (called collinear points).
In simple words: If the three points sit perfectly on the same straight path, they all belong to just one line. This happens when the points are collinear.

Exam Tip: Check if three points are collinear by seeing if you can place a ruler through all three without moving it - if yes, they form one line.

Question 12. Explain why is not possible for a line to have a mid point?
Answer: A line extends infinitely in both directions. So it is impossible to find its middle point. However, you can find the midpoint of a line segment, which has two defined endpoints.
In simple words: Since a line never ends and goes on forever, it has no middle. But a line segment has two ends, so you can find the exact middle between them.

Exam Tip: Always distinguish between a line (no endpoints, infinite length) and a line segment (two endpoints, finite length).

Question 13. Mark three non - collinear points points A, B, C in your notebook. Draw the lines through the points taking two at a time. Name these lines. How many such different lines can be drawn?
Answer: These are three non - collinear points A, B, C.

Three lines can be drawn through these points. These three lines are AB, BC, and AC.
In simple words: When you have three points that do not sit on the same line, you can connect them in pairs. Taking two at a time, you get three different line segments: one joining A and B, one joining B and C, and one joining A and C. These form the three sides of a triangle.

Exam Tip: Non-collinear points cannot all lie on one line - they form a triangle when you connect each pair, giving exactly three distinct line segments.

Question 14. Coplanar points are the points that are in the same plane. Thus,
(i) Can 150 points be coplanar?
(ii) Can 3 points be non - co planar?
Answer:
(i) Yes, a group of points that lie in the same plane are called coplanar points. Thus, it is possible that 150 points can be coplanar.
(ii) No, three points will be coplanar because we can have a plane that can contain 3 points on it. Thus, it is not possible that 3 points will be non - coplanar.
In simple words: Any number of points can fit on one flat surface, so 150 points can all be coplanar. But any three points always fit on some flat surface, so three points are always coplanar.

Exam Tip: Coplanar means all points fit on one plane - you need at least four points in special positions to be non-coplanar (3D space).

Question 15. Using a ruler, check whether the following points given in the figure are collinear or not?
Answer:
(i) D, A and C are collinear points
(ii) A, B and C are non - collinear points
(iii) A, B and E are collinear points
(iv) B, C and E are non - collinear points
In simple words: Collinear points all lie on one straight line. You can verify this by placing a ruler on the points - if the edge of the ruler touches all of them without moving the ruler, they are collinear. Otherwise, they are not.

Exam Tip: Use a straight edge like a ruler to test collinearity - it is faster and more accurate than trying to judge by eye alone.

Question 16. Lines p, q is coplanar. So are the lines p, r. Can we conclude that the lines p, q, r are coplanar?
Answer: No, p, q and r are not necessarily coplanar.
In simple words: Just because line p lies on one plane with line q, and p also lies on a plane with line r, does not mean all three lines fit on the same plane. The planes could be different, so all three might not be together.

Exam Tip: Being coplanar requires all three lines to share one common plane - pairwise coplanarity is not enough.

Question 17. Give three examples of each:
(i) Intersecting lines:
(ii) Parallel lines from your environment:
Answer:
(i) Intersecting lines - Two adjacent edges of your notebook, the letter X of the English alphabet, and crossing-roads.
(ii) Parallel lines from your environment - The cross-bars of a window, the opposite edges of a ruler (scale), and rail lines.
In simple words: Intersecting lines meet at a point, like roads crossing or the corners of pages. Parallel lines never meet and keep the same distance apart, like railway tracks or window panes.

Exam Tip: Always ensure your parallel line examples show lines that are truly equidistant - avoid examples where perspective makes them appear to meet in the distance.

Question 18. From the figure write:
(i) All pairs of intersecting lines - (l, m), (m, n) and (l, n)
(ii) All pairs of intersecting lines: (l, p), (m, p), (n, p), (l, r), (m, r), (n, r), (l, q), (m, q), (n, q), (q, p), (q, r)
(iii) Lines whose point of intersection is I: (m, p)
(iv) Lines whose point of intersection is D: (l, r)
(v) Lines whose point of intersection E: (m, r)
(vi) Lines whose point of intersection is A: (l, q)
(vii) Collinear points: (G, A, B and C), (D, E, J and F), (G, H, I and J, K), (A, H and D), (B, I and E) and (C, F and K)
In simple words: Read the diagram carefully. When two lines cross, note the point where they meet. Points that all sit on the same straight line are collinear.

Exam Tip: Label every intersection point clearly and double-check that all collinear point sets actually fall on one straight line in the figure.

Question 19. Write concurrent lines and their and their point of concurrence:
Answer: From the given figure, we have:

Concurrent lines can be defined as three or more lines which share the same meeting point. Clearly lines, n, q, and l are concurrent with A as the point of concurrence.

Lines, m, q and p are concurrent with B as the point of concurrence.
In simple words: Concurrent lines are three or more lines that all pass through one single point, called the point of concurrence. It is like having multiple roads meeting at one intersection.

Exam Tip: Be careful to identify all lines passing through each point - sometimes more than three lines meet at one spot.

Question 20. Mark four points A, B, C, D in your notebook such that no three of them are collinear. Draw all the lines which join them in pairs as shown
Answer: When you mark four points A, B, C, D such that no three lie on the same straight line, you can draw six different line segments by joining them in pairs: AB, AC, AD, BC, BD, and CD. These six lines connect every pair of points without any three points being collinear.
In simple words: With four points where no three are collinear, you get six lines. Take each point and draw lines to the other three points - that is 3 + 3 = 6 lines total (but you count each once). These form shapes like quadrilaterals or triangles with diagonals.

Exam Tip: To count the lines systematically: for n points in general position (no three collinear), the number of lines is n(n-1)/2.

Question 21. What is the maximum number of points of intersection of three lines in a plane? What is the minimum number?
Answer: The maximum number of points where three lines can intersect in a plane is three. This happens when no two lines are parallel and they all pass through different points. The minimum number of points where three lines can intersect is zero. This occurs when all three lines are parallel to each other.

Exam Tip: Remember that the maximum occurs when each pair of lines meets at a distinct point (no three lines meet at one point), and the minimum occurs when all lines are parallel or coincident.

Question 22. With the help of a figure, find the maximum and minimum number of points of intersection of four lines in a plane.
Answer: The maximum number of points where four lines can intersect in a plane is six. This happens when no two lines are parallel and no three lines meet at a single point. The minimum number of points where four lines can intersect is zero. This takes place when all four lines are parallel to each other.

Exam Tip: For n lines in general position (no two parallel, no three concurrent), the maximum intersection points equal \( \binom{n}{2} = \frac{n(n-1)}{2} \). For four lines: \( \frac{4 \times 3}{2} = 6 \) points.

Question 23. Lines p, q and r are concurrent. Also, the lines p, r and s are concurrent. Draw a figure and state whether lines p, q, r and s are concurrent or not?
Answer: When lines p, q, and r all meet at a common point O, they are concurrent at O. Similarly, if lines p, r, and s are concurrent, they too meet at a single point. Since both sets of lines share lines p and r, both groups meet at the same point O. Therefore, all four lines p, q, r, and s are concurrent at point O.

Exam Tip: If two different sets of lines share common lines and are individually concurrent, they must meet at the same point. This is because two distinct lines can intersect at only one point.

Question 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?
Answer: Since lines p, q, and r are concurrent, they intersect at a common point, say O. Since lines p, s, and t are concurrent, they also intersect at a common point. However, this second point need not be the same as O. Therefore, it is not always true that q, r, and s are concurrent, nor is it always true that q, r, and t are concurrent. The lines q, r, and s may intersect at different points, or they might not all pass through the same location.

Exam Tip: Concurrency of different groups of lines does not guarantee that other subsets of those lines will also be concurrent. Each group must be verified independently.

Question 25. Fill in the blanks in the following statements using suitable words:
(i) A page of a book is a physical example of a _____.
(ii) An inkpot has both _____surfaces.
(iii) Two lines in a plane are either _____ or are _____.
Answer:
(i) A page of a book is a physical example of a plane
(ii) An inkpot has both curved and plane surfaces
(iii) Two lines in a plane are either parallel or are intersecting
In simple words: A flat page shows what a plane looks like. A pen holder has curved parts and flat parts. Any two lines either run side by side without meeting or they cross each other.

Exam Tip: These are fundamental definitions in geometry. Make sure you learn what characterizes a plane (flat, extends infinitely), the types of surfaces (curved vs. plane), and the two possible relationships between lines in a plane (parallel or intersecting).

Question 26. State which of the following statements are true and which are false:
(i) Point has a size because we can see it as a thick dot on paper
(ii) By lines in geometry, we mean only straight lines
(iii) Two lines in a plane always intersect at a point
(iv) Any plane through a vertical line is vertical
(v) Any plane through a horizontal line is horizontal
(vi) There cannot be a horizontal line in a vertical plane
(vii) All lines in a horizontal plane are horizontal
(viii) Two lines in a plane always intersect at a plane
(ix) If two lines intersect at a point P, then P is called the point of c... the two lines
(x) If two lines intersect at a point P, then P is called the point of intersection of the two lines
(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
(xii) Two different lines can be drawn passing through two given points
(xiii) Through a given point only one line can be drawn
(xiv) Four points are collinear if any three of them lie on them lie on the same line
(xv) The maximum number of points of intersection of three lines is three
(xvi) The minimum number of points of intersection of three lines is one
Answer:
(i) False
(ii) True
(iii) False
(iv) True
(v) False
(vi) False
(vii) True
(viii) False
(ix) False
(x) True
(xi) False
(xii) False
(xiii) False
(xiv) False
(xv) True
(xvi) False
In simple words: A point is just a location with no size. Geometry deals with straight lines. Two lines may be parallel or intersect. Horizontal and vertical planes follow specific rules. One unique line passes through any two points.

Exam Tip: Pay close attention to the distinction between "always" and "sometimes" in these statements. Parallel lines never intersect, collinear points must all lie on one line, and through two distinct points exactly one line can be drawn.

Question 27. Give the correct matching of the statements of column A and column B

Exam Tip: These matching questions test your understanding of fundamental geometric definitions and properties. Commit each pairing to memory and be ready to explain the reasoning behind each match.

Exercise 10.2

Question 1. In the figure, points are given in two rows. Join the points AM, HE, TO, RUN, IF. How many line segments are formed?
Answer: When you join the points AM, HE, RUN, and IF, six line segments are created in total. These six line segments go by the names AM, HE, RU, IF, and UN.

Exam Tip: Count each distinct line segment carefully. A line segment has two endpoints, so identify which points form the endpoints of each segment.

Question 2. In the figure name:
(i) Five line segments
(ii) Five rays
(iii) non intersecting line segments
Answer:
(i) Five line segments - PQ, RS, PR, QS, AP
(ii) Five rays - QC->SD->PA->RB-> and RA
(iii) non intersecting line segments - PR, QS
In simple words: A line segment has two fixed endpoints. A ray starts at one point and extends forever in one direction. Non-intersecting segments never cross or touch each other.

Exam Tip: When naming rays, remember to include the direction arrow. For non-intersecting segments, check that they do not share any common points.

Question 3. In each of the following cases, state whether you can draw line segments on the given surfaces:
(i) The face of the cuboids
(ii) The surface of an egg or apple
(iii) The curved surface area of the cylinder. Four points such that there no three of them belong to the same line
(iv) The curved surface of the cone
(v) The base of the cone
Answer:
(i) Yes, you can draw line segments on the face of a cuboid because it is a flat plane surface.
(ii) No, you cannot draw a line segment on the curved surface of an egg or apple. Curved surfaces do not allow for true straight line segments to be drawn.
(iii) Yes, you can draw line segments on the curved surface of a cylinder. Any line segment that is parallel to the axis (the vertical centerline) of the cylinder will remain a line segment on that curved surface.
(iv) Yes, you can draw line segments on the curved surface of a cone. Any line segment joining the tip of the cone to a point on the base edge will form a line segment on the curved surface.
(v) Yes, you can draw line segments on the base of a cone because the base is flat. You can also draw line segments on the curved surface of the cone. Any line segment joining the tip of a cone to any point on the edge of the base will make a line segment.
In simple words: Flat surfaces let you draw straight line segments. Curved surfaces like eggs or apples do not. Some curved shapes like cones and cylinders do allow line segments along certain paths.

Exam Tip: Understand the difference between flat and curved surfaces. Cylinders allow segments parallel to their axis, and cones allow segments from apex to base circumference.

Question 4. Mark the following points on the sheet of the paper. Tell how many line segments can be obtained in the each case:
Answer: When you have n points on a plane and no three of them lie on the same straight line, the count of line segments you can form by joining these points equals \( \frac{n(n-1)}{2} \).

Using this formula:

(i) For two points A and B - number of line segments = \( 2(2-1) \div 2 = 1 \)

(ii) For three non-collinear points A, B, C - the count of line segments = \( 3(3-1) \div 2 = 3(2) \div 2 = 3 \)

(iii) When four points exist such that no three belong to the same line - the count of line segments = \( 4(4-1) \div 2 = 4(3) \div 2 = 6 \)

(iv) For any five points such that no three lie on the same line - the count of line segments = \( 5(5-1) \div 2 = 5(4) \div 2 = 10 \)
In simple words: To find how many line segments you can make from n points, multiply n by one less than n, then divide by 2. This works only if no three points sit on the same line.

Exam Tip: Remember the combinatorial formula \( \binom{n}{2} = \frac{n(n-1)}{2} \). This applies when you are choosing 2 points from n points to form a segment, and no three points are collinear.

Question 5. Count the number of line segments in figure
Answer: The line segments visible in the given figure are AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE. In total, there are 10 line segments present.

Exam Tip: When counting line segments systematically, list all segments starting from the first point, then move to the next point and list segments that have not yet been counted. This avoids duplication.

Question 6. In the figure name all the rays with initial points as A, B, C respectively.
Answer: The rays that begin at point A are: rays going toward P and toward Q in both directions along the line. Rays starting from point B would follow the same pattern based on the line structure. Rays with starting point C would similarly extend along the line in defined directions. A ray is named by its initial point followed by any point on the ray using arrow notation to show direction.

Exam Tip: Always include the arrow notation (like AP->) to show that a ray extends infinitely in one direction from its starting point. The order matters: the first letter is always the starting point.

Exercise 10.3

Question 1. Draw rough diagrams to illustrate the following:
(i) Open curve
(ii) Closed curve
Answer:
(i) Open curve: A spiral or arc shape that does not connect back to itself, with endpoints that remain separate from each other.
(ii) Closed curve: A shape like a square or circle where the line forms a continuous loop with no loose endpoints.
In simple words: An open curve has two free ends and never loops back on itself. A closed curve is fully enclosed and returns to where it started.

Exam Tip: Always ensure open curves have distinct endpoints while closed curves form complete, unbroken loops.

Question 2. Classify the following curves as open or closed?
(i) Open
(ii) Closed
(iii) Closed
(iv) Open
(v) Open
(vi) Closed
Answer: Each curve has been classified based on whether its endpoints meet or remain separate. Look at whether the line forms a complete loop (closed) or has free endpoints (open).
In simple words: If a curve's starting point and ending point are the same, it is closed. If they are different, the curve is open.

Exam Tip: Trace the curve with your finger - if you return to where you started without lifting your hand, it is closed; otherwise, it is open.

Question 3. Draw a polygon and shade its interior. Also draw its diagonals, if any
Answer: A polygon is a closed shape made entirely of straight line segments. ABCD is a four-sided polygon (quadrilateral). Its interior can be shaded. Diagonals are line segments connecting non-adjacent vertices - AC and BD are the two diagonals of this quadrilateral. These diagonals are drawn as straight lines from one corner to another corner that is not next to it.
In simple words: A polygon is a closed figure with straight sides. The inside can be shaded. Diagonals connect opposite corners, not the edges.

Exam Tip: Remember that diagonals go from one vertex to another vertex that is NOT directly adjacent - they skip at least one vertex.

Question 4. Illustrate, if possible, each one of the following with a rough diagram:
(i) A closed curve that is not a polygon
(ii) An open curve made up entirely of line segments
(iii) A polygon with two sides
Answer:
(i) A circle is a simple closed curve but not a polygon. A polygon must have line segments, while a circle is only a curve.
(ii) An open curve can be created by joining several line segments end to end without closing the shape. For example, a zigzag or bent line with free endpoints shows this.
(iii) A polygon with two sides is not possible. The minimum number of sides a polygon must have is three, forming a triangle.
In simple words: A circle closes but has no straight edges, so it is not a polygon. A zigzag line uses only straight pieces but stays open. You need at least three straight sides to make any polygon.

Exam Tip: Always recall that a polygon requires at least 3 sides and must be made entirely of straight line segments - these are the key definitions to distinguish valid shapes.