π Coordinate Geometry (Chapter 1) CBSE Class 9
Ganita Manjari | Complete Solutions with Detailed Explanation
π Exercise Set 1.1 (Based on Fig. 1.3)
Q1 Fig. 1.3 shows Reiaan’s room with points OABC marking its corners.
The x- and y-axes are marked in the figure.
Point O is the origin.
Referring to Fig. 1.3, answer the following questions:
The x- and y-axes are marked in the figure.
Point O is the origin.
Referring to Fig. 1.3, answer the following questions:
Q1(i) If D1R1 represents the door to Reiaan’s room, how far is the door from the left wall (the y-axis) of the room? How far is the door from the x-axis?
✅ Detailed Explanation: The door lies on the x-axis (horizontal line).
Distance from x-axis = 0 (since on axis).
From figure, D₁ = (8,0).
Distance from y-axis = x-coordinate = 8 units.
Left wall is the y-axis.
Distance from x-axis = 0 (since on axis).
From figure, D₁ = (8,0).
Distance from y-axis = x-coordinate = 8 units.
Left wall is the y-axis.
Q1(ii) What are the coordinates of D1?
Answer: D₁ = (8, 0).
It is 8 units right of origin on x-axis.
It is 8 units right of origin on x-axis.
Q1(iii) If R1 is the point (11.5, 0), how wide is the door? Do you think this is
a comfortable width for the room door? If a person in a wheelchair wants to enter the room, will he/she be able to do so easily?
Explanation: Width = |11.5 – 8| = 3.5 units.
Assuming 1 unit = 1 ft → 3.5 ft = 42 inches.
Standard room door: 30–36 inches.
42" is wider than average → comfortable.
Wheelchair needs ≥32 inches (≈2.7 ft).
Since 3.5 > 2.7,
yes, wheelchair accessible easily.
Assuming 1 unit = 1 ft → 3.5 ft = 42 inches.
Standard room door: 30–36 inches.
42" is wider than average → comfortable.
Wheelchair needs ≥32 inches (≈2.7 ft).
Since 3.5 > 2.7,
yes, wheelchair accessible easily.
Q1(iv) If B1 (0, 1.5) and B2 (0, 4) represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?
Solution: Bathroom width = 4 – 1.5 = 2.5 units.
Room door = 3.5 units.
2.5 is less than 3.5 → bathroom door is narrower
Room door = 3.5 units.
2.5 is less than 3.5 → bathroom door is narrower
π Think and Reflect (Real-world Observation)
π Question 1: What are the standard widths for a room door? Look around your home and in school.
✅ Detailed Answer:
| Location | Typical Width | Notes |
|---|---|---|
| Home (internal rooms) | 30–36 inches (2.5–3 ft) | Common in Indian homes |
| Home (main entrance) | 36–42 inches (3–3.5 ft) | Wider for furniture movement |
| School (classroom) | 36–48 inches (3–4 ft) | Must accommodate student flow |
| School (office/staff) | 30–36 inches (2.5–3 ft) | Standard internal door |
| International Standard (Wheelchair) | 32 inches (≈2.7 ft) minimum | ADA / Accessibility guidelines |
Conclusion: Standard room doors in Indian homes are typically 30–36 inches.
School doors are often wider at 36–48 inches.
International accessibility standards recommend a minimum of 32 inches clear width for wheelchair passage.
♿ Question 2: Are the doors in your school suitable for people in wheelchairs?
✅ Detailed Answer:
This depends on the specific school. Many modern schools in India now follow accessibility norms under the Rights of Persons with Disabilities Act (RPWD) 2016 and CBSE inclusive education guidelines.
✅ Suitable features (good schools):
- Door width ≥ 32 inches (2.7 ft) – allows wheelchair passage
- Ramps instead of steps at entrances
- Door handles at accessible height (not too high)
- Lightweight doors or automatic openers
- Thresholds that are flat or have small ramps
⚠️ Challenges in older schools:
- Doors may be only 24–28 inches wide – too narrow for wheelchairs
- Steps at entrance without ramps
- Heavy wooden doors difficult to open from a wheelchair
- Door handles placed too high
π Conclusion: While accessibility is improving across India, not all school doors are currently wheelchair suitable.
Students should observe their own school and suggest improvements if needed.
π Think and Reflect (Page 7 - Coordinate Concepts)
Q1 What is the x-coordinate of a point on the y-axis?
Answer: The x-coordinate is always 0.
Explanation: Any point on the y-axis has the form (0, y). Examples: (0, 5), (0, -3), (0, 0).
Explanation: Any point on the y-axis has the form (0, y). Examples: (0, 5), (0, -3), (0, 0).
Q2 Is there a similar generalisation for a point on the x-axis?
Answer: Yes. The y-coordinate of any point on the x-axis is always 0.
Explanation: Any point on the x-axis has the form (x, 0). Examples: (4, 0), (-7, 0).
Explanation: Any point on the x-axis has the form (x, 0). Examples: (4, 0), (-7, 0).
Q3 Does point Q (y, x) ever coincide with point P (x, y)? Justify your answer.
Answer: Yes, but only when x = y.
Justification: For two ordered pairs to be equal, their first coordinates must be equal AND their second coordinates must be equal. So x = y and y = x → both give x = y.
Example: If x = 3 and y = 3, then P(3,3) and Q(3,3) coincide. If x = 2 and y = 5, then P(2,5) and Q(5,2) are different points.
Justification: For two ordered pairs to be equal, their first coordinates must be equal AND their second coordinates must be equal. So x = y and y = x → both give x = y.
Example: If x = 3 and y = 3, then P(3,3) and Q(3,3) coincide. If x = 2 and y = 5, then P(2,5) and Q(5,2) are different points.
Q4 If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y. Is this claim true?
Answer: Yes, the claim is true.
Proof:
• If x ≠ y, then the ordered pairs differ in at least one coordinate → (x,y) ≠ (y,x).
• If (x,y) = (y,x), then equating first coordinates gives x = y.
• If x = y, then both pairs become (x,x) → they are equal.
Key insight: Ordered pairs are order-sensitive. The point (y,x) is generally the reflection of (x,y) across the line y = x.
Proof:
• If x ≠ y, then the ordered pairs differ in at least one coordinate → (x,y) ≠ (y,x).
• If (x,y) = (y,x), then equating first coordinates gives x = y.
• If x = y, then both pairs become (x,x) → they are equal.
Key insight: Ordered pairs are order-sensitive. The point (y,x) is generally the reflection of (x,y) across the line y = x.
✍️ Exercise Set 1.2 (Study table, bathroom, dining room)
Q1 On a graph sheet, mark the x-axis and y-axis and the origin O.
Mark points from (– 7, 0) to (13, 0) on the x-axis and from (0, – 15) to (0, 12) on the y-axis.
(Use the scale 1 cm = 1 unit.)
Using Fig. 1.5, answer the given questions
Mark points from (– 7, 0) to (13, 0) on the x-axis and from (0, – 15) to (0, 12) on the y-axis.
(Use the scale 1 cm = 1 unit.)
Using Fig. 1.5, answer the given questions
1(i) Place Reiaan’s rectangular study table with three of its feet at the
points (8, 9), (11, 9) and (11, 7). (i)Where will the fourth foot of the table be?
Reason: Rectangle: x-coordinate same as A(8) and y same as C(7) → (8,7).
1(ii) (ii) Is this a good spot for the table?
Answer: Yes, placed against the wall, does not block movement.
1(iii) What is the width of the table? The length? Can you make out
the height of the table?
Width = 11-8 = 3 units; Length = 9-7 = 2 units. Height cannot be known (2D top view only).
2 If the bathroom door has a hinge at B1
and opens into the bedroom,
will it hit the wardrobe? Are there any changes you would suggest
if the door is made wider?
Door width = 2.5 units. Wardrobe starts at x=3 (>2.5) → will not hit. If made wider, shift wardrobe or make door open inward.
3(i) Look at Reiaan’s bathroom.
(i) What are the coordinates of the four corners O, F, R, and P of the bathroom?
From Fig 1.5: O(0,9), F(0,0), R(-6,9), P(-6,0).
3(ii) (ii) What is the shape of the showering area SHWR in Reiaan’s
bathroom? Write the coordinates of the four corners.
Shape = Trapezium. S(-6,5), H(-3,5), W(-2,9), R(-6,9).
3(iii) (iii) Mark off a 3 ft × 2 ft space for the washbasin and a 2 ft × 3 ft
space for the toilet. Write the coordinates of the corners of these spaces.
Washbasin: (-6,0), (-3,0), (-3,2), (-6,2). Toilet: (-6,2), (-4,2), (-4,5), (-6,5).
4(i) 4. Other rooms in the house:
(i)
Reiaan’s room door leads from the dining room which has
the length 18 ft and width 15 ft. The length of the dining room
extends from point P to point A. Sketch the dining room and
mark the coordinates of its corners.
P(-6,0), A(12,0), Q(12,-15), S(-6,-15).
4(ii) (ii)
Place a rectangular 5 ft × 3 ft dining table precisely in the
centre of the dining room. Write down the coordinates of the
feet of the table.
Centre (3, -7.5). Feet: (0.5,-9), (5.5,-9), (5.5,-6), (0.5,-6).
π Think and Reflect (Page 9 - Distance using Baudhayana-Pythagoras)
Q1 In moving from A (3, 4) to D (7, 1), what distance has been covered
along the x-axis? What about the distance along the y-axis?
Answer:
• Distance along x-axis: 7 − 3 = 4 units (moving right)
• Distance along y-axis: 4 − 1 = 3 units (moving down)
Note: We take absolute values, so direction doesn't matter for distance.
• Distance along x-axis: 7 − 3 = 4 units (moving right)
• Distance along y-axis: 4 − 1 = 3 units (moving down)
Note: We take absolute values, so direction doesn't matter for distance.
Q2 Can these distances help you find the distance AD?
Answer: Yes! Using the Baudhayana-Pythagoras Theorem:
AD = √[(x-distance)² + (y-distance)²] = √[(4)² + (3)²] = √[16 + 9] = √25 = 5 units.
The horizontal and vertical distances form the legs of a right triangle, and AD is the hypotenuse.
AD = √[(x-distance)² + (y-distance)²] = √[(4)² + (3)²] = √[16 + 9] = √25 = 5 units.
The horizontal and vertical distances form the legs of a right triangle, and AD is the hypotenuse.
π Think and Reflect (Page 11 - Reflection in Axes)
Q1 What has remained the same and what has changed with this reflection?
Answer:
✓ Remained the same: y-coordinates, side lengths (AD, DM, MA), shape, size, area, congruence.
✗ Changed: x-coordinates change sign (positive ↔ negative), triangle flips horizontally, position/quadrant changes.
✓ Remained the same: y-coordinates, side lengths (AD, DM, MA), shape, size, area, congruence.
✗ Changed: x-coordinates change sign (positive ↔ negative), triangle flips horizontally, position/quadrant changes.
Q2 Would these observations be the same if ΞADM is reflected in the x-axis (instead of the y-axis)?
Answer: Yes, the observations are similar, but the roles swap:
• Reflection in y-axis: x-coordinates change sign; y-coordinates unchanged.
• Reflection in x-axis: y-coordinates change sign; x-coordinates unchanged.
• In both cases: side lengths, shape, size, area, and congruence are preserved.
• Reflection in y-axis: x-coordinates change sign; y-coordinates unchanged.
• Reflection in x-axis: y-coordinates change sign; x-coordinates unchanged.
• In both cases: side lengths, shape, size, area, and congruence are preserved.
π End-of-Chapter Exercises (Full Solutions)
1. What are the x-coordinate and y-coordinate of the point of intersection of the two axes?
Origin (0,0).
2. Point W has x-coordinate equal to – 5. Can you predict the coordinates of point H which is on the line through W parallel to the y-axis? Which quadrants can H lie in?
H = (-5, y). QII if (y>0), QIII if (y<0), on axis if y=0.
3. Consider the points R (3, 0), A (0, – 2), M (– 5, – 2) and P (– 5, 2). If they are joined in the same order, predict:
(i) Two sides of RAMP that are perpendicular to each other.
(ii) One side of RAMP that is parallel to one of the axes.
(iii) Two points that are mirror images of each other in one axis.
Which axis will this be?
Now plot the points and verify your predictions.
Perpendicular: AM ⟂ MP. Parallel to axis: AM ∥ x-axis. Mirror: M and P about x-axis.
4. Plot point Z (5, – 6) on the Cartesian plane. Construct a right-angled triangle IZN and find the lengths of the three sides.
(Comment: Answers may differ from person to person.)
(Activity) Example: I(5,0) → IZ=6, ZN depends on N. Students verify.
5.What would a system of coordinates be like if we did not have negative numbers? Would this system allow us to locate all the points on a 2-D plane?
Only Quadrant I and positive axes. Cannot locate QII, QIII, QIV.
6. Are the points M (– 3, – 4), A (0, 0) and G (6, 8) on the same straight line? Suggest a method to check this without plotting and joining the points.
MA=5, AG=10, MG=15 → MA+AG=MG → collinear.
7. Use your method (from Problem 6) to check if the points
R (– 5, – 1), B (– 2, – 5) and C (4, – 12) are on the same straight line.
Now plot both sets of points and check your answers.
RB=5, BC=√85≈9.22, RC=√202≈14.21 → 5+9.22≠14.21 → not collinear.
8(i)Using the origin as one vertex, plot the vertices of:(i) A right-angled isosceles triangle.
O(0,0), A(4,0), B(0,4).
8(ii) (ii) An isosceles triangle with one vertex in Quadrant III and the other in Quadrant IV
O(0,0), P(-3,-4), Q(3,-4). OP=OQ=5.
9. Midpoint table (S,M,T).
(-3,0)(0,0)(3,0)→Yes; (2,3)(3,4)(4,5)→Yes; (0,0)(0,5)(0,-10)→No; (-8,7)(0,-2)(6,-3)→No.
10. M(-7,1) midpoint of A(3,-4) and B(x,y). Find B.
x = -17, y = 6 → B(-17,6).
11. Trisection of A(4,7), B(16,-2): P (near A), Q.
P(8,4), Q(12,1).
12. Circle K centre O(0,0), A(1,-8), B(-4,7), C(-7,-4). Radius? D,E inside/outside?
Radius = √65 ≈8.06. D(-5,6): √61 < √65 → inside. E(0,9): 9 > √65 → outside.
13. Midpoints D(5,1),E(6,5),F(0,3) of ΞABC. Find A,B,C.
A(1,-1), B(11,3), C(-5,7).
14. City grid: intersections (4,3) and (3,4)?
Both refer to exactly 1 unique crossing each.
15. Computer screen 800×600. Circles A(100,150) r=80; B(250,230) r=100. Outside? Intersect?
Both circles fully inside (check edges). Distance AB = 170; sum radii =180 → 170<180 → intersect.
16. A(2,1), B(-1,2), C(-2,-1), D(1,-2). Square? Area?
All sides = √10, diagonals = √20 → square. Area = 10 sq units.
π Chapter Summary (as per Ganita Manjari)
- Two perpendicular lines: x-axis (horizontal) and y-axis (vertical).
- Origin = (0,0). Quadrants: I(+,+), II(-,+), III(-,-), IV(+,-).
- Distance formula: √[(x₂−x₁)² + (y₂−y₁)²] (Baudhayana-Pythagoras).
- Midpoint formula: ((x₁+x₂)/2 , (y₁+y₂)/2).
- If x=y then (x,y)=(y,x); otherwise ordered pair matters.
