📘 About the new NCERT Class 9 book Ganita Manjari
The new NCERT Maths book “Ganita Manjari” ✨ is designed to make mathematics more interactive, meaningful, and connected to real life 🧠🌍. Instead of only focusing on formulas, it:
- Uses stories and situations (like Reiaan’s room 🏠) to explain concepts
- Encourages thinking, reasoning, and exploration 🤔
- Includes “Think and Reflect” questions 💭 to build deeper understanding
- Connects maths with history and practical use 📜📏
👉 Overall, this new book focuses less on rote learning and more on understanding concepts clearly and applying them in real situations 🌐.
Exercise Set 1.1 – Coordinate Geometry Questions and Answers
Fig. 1.3 shows Reiaan’s room with points OABC marking its corners. The x-axis and y-axis are shown in the figure, and point O represents the origin.
Referring to Fig. 1.3, answer the following questions.
(i) Distance of the Door from the Walls
If D₁R₁ represents the door to Reiaan’s room:
-
How far is the door from the left wall (y-axis)?
-
How far is the door from the x-axis?
Answer
The room door lies on the x-axis. Therefore, its distance from the x-axis is:
0 units
From the figure:
-
D₁ = (8, 0)
-
R₁ = (11.5, 0)
Hence, the door starts 8 units away from the y-axis.
(ii) Coordinates of D₁
Answer
The coordinates of point D₁ are:
(8, 0)
(iii) Width of the Door and Wheelchair Accessibility
If R₁ is the point (11.5, 0), find the width of the door. Is it comfortable for regular use and wheelchair accessibility?
Answer
Given:
-
D₁ = (8, 0)
-
R₁ = (11.5, 0)
Width of the door:
[
11.5 – 8 = 3.5 \text{ units}
]
So, the width of the door is:
3.5 units
Is This Width Comfortable?
-
If 1 unit represents 1 foot, then:
3.5 feet = 42 inches
-
Standard residential doors are usually 30–36 inches wide.
-
Therefore, this door is slightly wider than average and provides comfortable access.
Wheelchair Accessibility
A wheelchair generally requires a minimum clear width of:
32 inches (about 2.7 feet)
Since:
[
3.5 \text{ ft} > 2.7 \text{ ft}
]
the door is wide enough for wheelchair users.
Conclusion
Yes, the door width is comfortable, and a person using a wheelchair should be able to enter easily.
(iv) Comparison of Bathroom Door and Room Door
If:
-
B₁ = (0, 1.5)
-
B₂ = (0, 4)
represent the ends of the bathroom door, determine whether the bathroom door is narrower or wider than the room door.
Answer
Width of the bathroom door:
[
4 – 1.5 = 2.5 \text{ units}
]
Width of the room door:
[
3.5 \text{ units}
]
Since:
[
2.5 < 3.5
]
the bathroom door is narrower than the room door.
Exercise Set 1.2 – Coordinate Geometry Solutions
Draw the x-axis and y-axis on a graph sheet and mark the origin O. Plot points from:
-
(-7, 0) to (13, 0) on the x-axis
-
(0, –15) to (0, 12) on the y-axis
Use the scale:
1 cm = 1 unit
Using Fig. 1.5, answer the following questions.
Study Table Placement Questions and Answers
Reiaan’s rectangular study table has three feet placed at:
-
(8, 9)
-
(11, 9)
-
(11, 7)
(i) Coordinates of the Fourth Foot
Answer
The given points form three corners of a rectangle:
-
A = (8, 9)
-
B = (11, 9)
-
C = (11, 7)
To complete the rectangle, the fourth corner must have:
-
the same x-coordinate as A → 8
-
the same y-coordinate as C → 7
Therefore, the coordinates of the fourth foot are:
(8, 7)
(ii) Is This a Good Position for the Table?
Answer
Yes, this is a suitable location for the study table because:
-
The table is properly placed inside the room.
-
It does not block doors or walking paths.
-
It is positioned close to the wall, making it practical for studying and saving space.
(iii) Width, Length, and Height of the Table
Answer
Width of the Table
Distance between:
-
(8, 9) and (11, 9)
[
11 – 8 = 3 \text{ units}
]
So, the width is:
3 units
Length of the Table
Distance between:
-
(11, 9) and (11, 7)
[
9 – 7 = 2 \text{ units}
]
So, the length is:
2 units
Height of the Table
The height cannot be determined because the figure shows only a 2-dimensional top view of the room.
Bathroom Door and Wardrobe Question
If the bathroom door has a hinge at B₁ and opens into the bedroom, will it hit the wardrobe? Suggest changes if the door is made wider.
Answer
Given:
-
B₁ = (0, 1.5)
-
B₂ = (0, 4)
Width of the bathroom door:
[
4 – 1.5 = 2.5 \text{ units}
]
The door rotates in an arc of radius:
2.5 units
The wardrobe begins near:
-
W₁ = (3, 0)
-
W₄ = (3, 2)
Since the wardrobe is farther than 2.5 units from the hinge point, the bathroom door will not hit the wardrobe.
Suggested Changes if the Door is Wider
If the door becomes wider, it may interfere with the wardrobe. In that case:
-
the door could open inward into the bathroom,
-
the wardrobe could be shifted slightly,
-
or the door width could be reduced for smooth movement.
Reiaan’s Bathroom – Coordinate Geometry Questions
(i) Coordinates of Bathroom Corners
Find the coordinates of O, F, R, and P.
Answer
-
O = (0, 0)
-
F = (0, 9)
-
R = (-6, 9)
-
P = (-6, 0)
(ii) Shape and Coordinates of SHWR
Find the shape of SHWR and write the coordinates of its corners.
Answer
Coordinates:
-
S = (-6, 5)
-
H = (-3, 5)
-
W = (-2, 9)
-
R = (-6, 9)
Since one pair of opposite sides is parallel, SHWR is a:
Trapezium
Coordinates of SHWR
-
S = (-6, 5)
-
H = (-3, 5)
-
W = (-2, 9)
-
R = (-6, 9)
(iii) Coordinates for Washbasin and Toilet Spaces
Washbasin Space (3 ft × 2 ft)
Coordinates:
-
(-6, 0)
-
(-3, 0)
-
(-3, 2)
-
(-6, 2)
Toilet Space (2 ft × 3 ft)
Coordinates:
-
(-6, 2)
-
(-4, 2)
-
(-4, 5)
-
(-6, 5)
-
Other Rooms in the House – Coordinate Geometry Solutions
(i) Coordinates of the Dining Room Corners
Reiaan’s room door opens into the dining room. The dining room has:
-
Length = 18 ft
-
Width = 15 ft
The length extends from point P to point A.
Answer
Given:
-
P = (−6, 0)
-
A = (12, 0)
Length of PA:
[
12 – (-6) = 18 \text{ ft}
]Since the dining room is 15 ft wide and extends downward from PA, the y-coordinate decreases by 15 units.
Therefore, the coordinates of the four corners are:
-
P = (−6, 0)
-
A = (12, 0)
-
Q = (12, −15)
-
S = (−6, −15)
Coordinates of the Dining Room
[
(-6,0),\ (12,0),\ (12,-15),\ (-6,-15)
]
(ii) Coordinates of the Dining Table Feet
A rectangular dining table of size 5 ft × 3 ft is placed exactly at the centre of the dining room.
Answer
The dining room extends:
-
from x = −6 to x = 12
-
from y = 0 to y = −15
Centre of the Dining Room
x-coordinate of centre:
[
\frac{-6 + 12}{2} = 3
]y-coordinate of centre:
[
\frac{0 + (-15)}{2} = -7.5
]So, the centre point is:
[
(3,\ -7.5)
]Dimensions of the Table
-
Length = 5 ft → half-length = 2.5 ft
-
Width = 3 ft → half-width = 1.5 ft
Coordinates of the Table Corners
-
(0.5, −9)
-
(5.5, −9)
-
(5.5, −6)
-
(0.5, −6)
End-of-Chapter Exercises – Coordinate Geometry Answers
Intersection Point of the Axes
What are the x-coordinate and y-coordinate of the point where the two axes intersect?
Answer
The x-axis and y-axis intersect at the origin.
Coordinates of the origin:
[
(0,\ 0)
]So:
-
x-coordinate = 0
-
y-coordinate = 0
Coordinates of Point H on a Vertical Line
Point W has x-coordinate equal to −5. Predict the coordinates of point H on the line through W parallel to the y-axis.
Answer
Any point on a line parallel to the y-axis has the same x-coordinate.
Therefore:
[
H = (-5,\ y)
]Possible Quadrants
-
If ( y > 0 ), H lies in Quadrant II
-
If ( y < 0 ), H lies in Quadrant III
-
If ( y = 0 ), H lies on the x-axis
Properties of Quadrilateral RAMP
Given points:
-
R = (3, 0)
-
A = (0, −2)
-
M = (−5, −2)
-
P = (−5, 2)
joined in the same order.
(i) Perpendicular Sides of RAMP
Answer
-
AM is horizontal
-
MP is vertical
Therefore:
[
AM \perp MP
]
(ii) Side Parallel to an Axis
Answer
-
AM is parallel to the x-axis
-
MP is parallel to the y-axis
(iii) Mirror Image Points
Answer
Points:
-
M = (−5, −2)
-
P = (−5, 2)
have:
-
same x-coordinate
-
opposite y-values
Therefore, M and P are mirror images of each other in the x-axis.
Right-Angled Triangle on the Cartesian Plane
Plot point Z(5, −6) and construct a right-angled triangle IZN.
Answer
Take:
-
I = (5, 0)
-
N = (0, −6)
Lengths of the Sides
IZ
[
IZ = 6 \text{ units}
]ZN
[
ZN = 5 \text{ units}
]IN
Using the distance formula:
IN=\sqrt{(5-0)^2+(0-(-6))^2}=\sqrt{25+36}=\sqrt{61}
Therefore:
-
IZ = 6 units
-
ZN = 5 units
-
IN = √61 units
Importance of Negative Numbers in Coordinate Geometry
What would happen if negative numbers did not exist in the coordinate system?
Answer
Without negative numbers:
-
only positive coordinates and zero could be represented
-
only Quadrant I would exist
We would not be able to represent:
-
Quadrant II
-
Quadrant III
-
Quadrant IV
-
negative portions of the axes
Therefore, many points on the Cartesian plane could not be located.
Collinearity of Points M, A, and G
Check whether:
-
M = (−3, −4)
-
A = (0, 0)
-
G = (6, 8)
lie on the same straight line.
Answer
Using the distance formula:
-
MA = 5
-
AG = 10
-
MG = 15
Since:
[
MA + AG = MG
]the three points lie on the same straight line.
Check Whether Points R, B, and C are Collinear
Given:
-
R = (−5, −1)
-
B = (−2, −5)
-
C = (4, −12)
Answer
Using the distance formula:
-
RB = 5
-
BC = √85
-
RC = √202
Since:
[
RB + BC \ne RC
]the points do not lie on the same straight line.
Triangles Using the Origin as One Vertex
(i) Right-Angled Isosceles Triangle
Answer
Take:
-
O = (0, 0)
-
A = (4, 0)
-
B = (0, 4)
Then:
-
OA = 4
-
OB = 4
-
OA ⟂ OB
Therefore, triangle OAB is a right-angled isosceles triangle.
(ii) Isosceles Triangle in Quadrants III and IV
Answer
Take:
-
O = (0, 0)
-
P = (−3, −4)
-
Q = (3, −4)
Then:
-
OP = 5
-
OQ = 5
Therefore, triangle OPQ is an isosceles triangle.
Midpoint Verification Questions
Determine whether M is the midpoint of ST.
Row 1
-
S = (−3, 0)
-
M = (0, 0)
-
T = (3, 0)
Answer
-
SM = 3
-
MT = 3
Since:
[
SM = MT
]M is the midpoint of ST.
Row 2
-
S = (2, 3)
-
M = (3, 4)
-
T = (4, 5)
Answer
-
SM = √2
-
MT = √2
Since:
[
SM = MT
]M is the midpoint of ST.
Row 3
-
S = (0, 0)
-
M = (0, 5)
-
T = (0, −10)
Answer
-
SM = 5
-
MT = 15
Since:
[
SM \ne MT
]M is not the midpoint of ST.
-
Coordinate Geometry Questions and Answers – Advanced Exercises
Row 4 – Checking Midpoint
Given:
-
S = (−8, 7)
-
M = (0, −2)
-
T = (6, −3)
Answer
Using the distance formula:
[
SM = \sqrt{145}
]
[
MT = \sqrt{37}
]
Since:
[
SM \ne MT
]
point M is not the midpoint of ST.
Finding Coordinates Using the Midpoint Formula
Given:
-
A = (3, −4)
-
M = (−7, 1)
where M is the midpoint of AB, find the coordinates of B(x, y).
Answer
Using the midpoint formula:
\left(\frac{3+x}{2},\frac{-4+y}{2}\right)=(-7,1)
From the x-coordinates
[
\frac{3 + x}{2} = -7
]
[
3 + x = -14
]
[
x = -17
]
From the y-coordinates
[
\frac{-4 + y}{2} = 1
]
[
-4 + y = 2
]
[
y = 6
]
Therefore, the coordinates of B are:
[
(-17,\ 6)
]
Coordinates of Trisection Points
Let P and Q divide line segment AB into three equal parts, where:
-
P is closer to A
-
Q is closer to B
Given:
-
A = (4, 7)
-
B = (16, −2)
Answer
Point P divides AB in the ratio 1 : 2.
Point Q divides AB in the ratio 2 : 1.
Therefore:
[
P = (8,\ 4)
]
[
Q = (12,\ 1)
]
Points on a Circle Centred at the Origin
(i) Show That Points Lie on a Circle
Given:
-
A = (1, −8)
-
B = (−4, 7)
-
C = (−7, −4)
Answer
Distance from origin to each point:
[
OA = \sqrt{65}
]
[
OB = \sqrt{65}
]
[
OC = \sqrt{65}
]
Since all three distances are equal, the points lie on the same circle centred at the origin.
Radius of the Circle
[
\sqrt{65}
]
(ii) Points Inside or Outside the Circle
Given:
-
D = (−5, 6)
-
E = (0, 9)
Answer
Distance from origin:
[
OD = \sqrt{61}
]
[
OE = 9
]
Radius of the circle:
[
\sqrt{65} \approx 8.06
]
Position of D
Since:
[
\sqrt{61} < \sqrt{65}
]
point D lies inside the circle.
Position of E
Since:
[
9 > \sqrt{65}
]
point E lies outside the circle.
Finding Vertices of a Triangle from Midpoints
The midpoints of triangle ABC are:
-
D = (5, 1)
-
E = (6, 5)
-
F = (0, 3)
Find the coordinates of A, B, and C.
Answer
Using the midpoint formula, the coordinates are:
[
A = (1,\ 7)
]
[
B = (-1,\ -1)
]
[
C = (11,\ 3)
]
City Street Coordinate Model
A city has two main roads intersecting at the centre. All other roads are parallel and 200 m apart. There are 10 streets in each direction.
(i) Drawing the Model
Answer
Using the scale:
[
1 \text{ cm} = 200 \text{ m}
]
draw:
-
10 vertical parallel lines
-
10 horizontal parallel lines
with each line spaced 1 cm apart.
(ii) Street Intersections
(a) Intersections Named (4, 3)
Answer
Only one intersection can be referred to as:
[
(4,\ 3)
]
(b) Intersections Named (3, 4)
Answer
Only one intersection can be referred to as:
[
(3,\ 4)
]
Computer Graphics and Coordinate Geometry
A computer screen has dimensions:
[
800 \times 600 \text{ pixels}
]
Two circles are displayed:
Circle 1
-
Centre A = (100, 150)
-
Radius = 80
Circle 2
-
Centre B = (250, 230)
-
Radius = 100
(i) Do Any Circles Lie Outside the Screen?
Answer
No. Both circles lie completely inside the screen boundaries.
(ii) Do the Circles Intersect?
Answer
Yes, the two circles intersect each other.
Checking Whether ABCD is a Square
Given points:
-
A = (2, 1)
-
B = (−1, 2)
-
C = (−2, −1)
-
D = (1, −2)
Determine whether ABCD is a square and find its area.
Answer
Using the distance formula:
[
AB = \sqrt{10}
]
[
BC = \sqrt{10}
]
[
CD = \sqrt{10}
]
[
DA = \sqrt{10}
]
Since all four sides are equal, ABCD is a rhombus.
Diagonals
[
AC = \sqrt{20}
]
[
BD = \sqrt{20}
]
Since the diagonals are equal, ABCD is a square.
Area of the Square
[
\text{Area} = (\sqrt{10})^2
]
[
= 10 \text{ square units}
]
Therefore, the area of square ABCD is:
[
10 \text{ square units}
]