Orienting Yourself: The Use of Coordinates NCERT Solutions Ch 1 Class 9

Master the fundamentals of coordinate geometry with our comprehensive Orienting Yourself: The Use of Coordinates NCERT Solutions. Specifically designed for the CBSE curriculum, these solutions align perfectly with the Ganita Manjari | Grade 9 | Part I textbook.

Whether you are tackling Exercise Set 1.1, Exercise Set 1.2, or End-of-Chapter Exercises, our step-by-step guide ensures you grasp every concept of the NCERT syllabus with ease.

Only the answers are given first, and then detailed solutions are provided

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Exercise Set 1.1 – Answers

Exercise Set 1.1 Step-by-Step Solutions Class 9 Math NCERT New Book

Q (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?

Solution:

Point $D_1$D1​ lies on the x-axis at approximately $(10, 0)$

Distance from the y-axis is equal to the x-coordinate (distance from y-axis = x-coordinate) → $= 10$ units

Distance from the x-axis is equal to the y-coordinate
(distance from x-axis = y-coordinate) → $= 0$ units

Final Answer:

• Distance from y-axis = 10 units
• Distance from x-axis = 0 units

Q(ii) What are the coordinates of D1?

Point $D_1$​ lies on the x-axis, so its y-coordinate is 0
(points on the x-axis have coordinates (x, 0).

From the figure, its x-coordinate is 10.$$D_1 = (10, 0)$$Final Answer:

(10, 0)

Q(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?

Solution:

Both $D_1$​ and $R_1$ lie on the x-axis, so the width is the difference of their x-coordinates
(distance on the x-axis = difference of x-values).
$$\text{Width} = 11.5 – 10 = 1.5 \text{ units}$$

A standard door is about 3 feet wide, so 1.5 units is quite small.

Hence, the door is not comfortable and is not suitable for a wheelchair.

Final Answer:

● Width = 1.5 units
● Not comfortable
● Not wheelchair-friendly

Q(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:

Both points lie on the y-axis, so the width is the difference of their y-coordinates
(distance on the y-axis = difference of y-values).
$$\text{Bathroom door width} = 4 – 1.5 = 2.5 \text{ units}$$

Room door width = 1.5 units

Since $2.5 > 1.5$, the bathroom door is wider.

Final Answer:

The bathroom door is wider than the room door

Exercise Set 1.2 – Answers

Detailed solutions Exercise Set 1.2

1. 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?
   (ii) Is this a good spot for the table?
   (iii) What is the width of the table? The length? Can you make out
   The height of the table?

Solution:

(i) The three given points are $(8, 9), (11, 9), (11, 7)$(8,9),(11,9),(11,7).

These form two perpendicular sides of a rectangle

[In a rectangle, adjacent sides are perpendicular and opposite sides are parallel to the axes here.]

The fourth point must have the x-coordinate of the first point and the y-coordinate of the third point

(completing the rectangle by matching x and y values):

(8,7) Ans.

(ii) The placement is suitable as the table aligns neatly without blocking space
(based on spatial arrangement in Fig. 1.5).

(iii)
Width is the horizontal distance:
$$11 – 8 = 3 \text{ units} \quad (\text{difference of x-coordinates})$$

Length is the vertical distance:
$$9 – 7 = 2 \text{ units} \quad (\text{difference of y-coordinates})$$

Height cannot be determined because only a 2D view is provided.

[ The coordinates represent only length and breadth.]

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?

Solution:

The door rotates about point $B_1$ (door motion is circular about the hinge).
From Fig. 1.5, the wardrobe lies close to the path of the door.

If the door opens inward, its arc may intersect the wardrobe area (as rotation increases, the sweep area increases).

Making the door wider increases this arc further.

So, it may hit the wardrobe. A better design would be to change the hinge position or make the door open outward.

3. Look at Reiaan’s bathroom.
   (i) What are the coordinates of the four corners O, F, R, and P of
   the bathroom?
   (ii) What is the shape of the showering area SHWR in Reiaan’s
   bathroom? Write the coordinates of the four corners.
   (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

Solution:

(i) From Fig. 1.5, the bathroom corners are read directly:
$$O (0, 0), \; F (0, 10), \; R (-7, 10), \; P (-7, 0)$$

(ii) The shower area has opposite sides parallel and equal with right angles (rectangle properties), so it is a rectangle.

Approximate coordinates (from diagram):
$$(-6, 8), (-2, 8), (-2, 5), (-6, 5)$$

(iii)

For washbasin (3 × 2):
Choose any suitable rectangular region of size 3 units by 2 units (length and breadth fixed), e.g:
$$(-6, 2), (-3, 2), (-3, 4), (-6, 4)$$

For toilet (2 × 3):$$(-6, 5), (-4, 5), (-4, 8), (-6, 8)$$

4. Other rooms in the house:
   (i) Reiaan’s room door leads from the dining room, which has
   a length of 18 ft and a width of 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.
   (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.

Solution:

(i) Assume coordinates consistent with given dimensions
(using coordinate system scaling):

Dining room corners can be taken as:
$$(0,0), (18,0), (18,15), (0,15)$$

(ii) The centre of the rectangle is:
$$\left(\frac{0+18}{2}, \frac{0+15}{2}\right) = (9, 7.5) \quad (\text{midpoint formula})$$

Table dimensions are 5 × 3,

So half-lengths are 2.5 and 1.5.

Coordinates of table corners:
$$(6.5, 6), (11.5, 6), (11.5, 9), (6.5, 9)$$

End-of-Chapter Exercises
Page 12 to 13

Q1. What are the x- and y-coordinates of the point of
Intersection of the two axes?

Solution:

• The x-axis and y-axis intersect at the origin
• From the concept: origin = (0, 0)

Final Answer:

(0, 0)

Q2.Point W has an 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?

Solution:

• Line parallel to the y-axis → x remains constant
• So H = (–5, y)
• Depending on y:
  • y > 0 → Quadrant II
  • y < 0 → Quadrant III

Final Answer:

H = (–5, y); Quadrants II and III

Q3. 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.

Solution:

Step 1: Perpendicular sides

• Horizontal vs vertical sides
• AM (horizontal), MP (vertical)

Step 2: Parallel side

• AM lies parallel to the x-axis

Step 3: Mirror image

• M (–5, –2) and P (–5, 2)
• Reflection in the x-axis

Final Answer:

(i) AM ⟂ MP
(ii) AM ∥ x-axis
(iii) M and P; x-axis

Q4. 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.)

Solution:

Take I = (5, 0), N = (0, –6)

Using distance on axes:

I Z = 6 and Z N = 5

Using Pythagoras:

Final Answer:

I Z = 6 Z N = 5 IN = √61

Q5. 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?

Solution:

• Only the first quadrant would exist
• Cannot represent left or downward points

Final Answer:

No, only first-quadrant points are possible

Q6. 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.

Solution:

Check slope:

• MA slope = 4/3
• AG slope = 8/6 = 4/3

Equal slopes → collinear

Final Answer:

Yes, they lie on a straight line

Q7. 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.

Solution:

• RB slope = –4/3
• BC slope = –7/6

Not equal → not collinear

Final Answer:

No, they are not collinear

Q8. Using the origin as one vertex, plot the vertices of:
(i) A right-angled isosceles triangle.
(ii) An isosceles triangle with one vertex in Quadrant III and the other
other in Quadrant IV.

Solution:

(i) Right isosceles triangle:

Example: (0,0), (2,0), (0,2) Ans

(ii) Isosceles triangle (QIII & QIV):

Example: (–2, –2), (2, –2), (0, –5) Ans

Q9. The following table shows the coordinates of points S, M, and
T. In each case, state whether M is the midpoint of the segment
ST. Justify your answer

When M is the mid-point of ST, can you find any connection
between the coordinates of M, S and T?

Solution:

Use the midpoint formula:

Q10. Use the connection you found to find the coordinates of B, given
that M (–7, 1) is the midpoint of A (3, – 4) and B (x, y).

Solution:

Using midpoint formula:$$(-7,1) = \left(\frac{3+x}{2}, \frac{-4+y}{2}\right)$$(−7,1)=(23+x​,2−4+y​)

Solve:
x = –17, y = 6

Final Answer:

B = (–17, 6)

Q11. Let P, Q be points of trisection of AB, with P closer to A, and Q
closer to B. Using your knowledge of how to find the coordinates
of the midpoint of a segment, how would you find the coordinates
of P and Q? Do this for the case when the points are A (4, 7) and
B (16, –2).

Solution:

$$P = \left(\frac{2x_1+x_2}{3}, \frac{2y_1+y_2}{3}\right)$$$$Q = \left(\frac{x_1+2x_2}{3}, \frac{y_1+2y_2}{3}\right)$$

Calculate:
P = (8, 4)
Q = (12, 1)

Final Answer:

P = (8, 4), Q = (12, 1)

Q12. (i) Given the points A (1, 8), B (– 4, 7), and C (–7, – 4), show that
They lie on a circle K whose centre is the origin O (0, 0). What
Is the radius of circle K?
(ii) Given the points D (– 5, 6) and E (0, 9), check whether D and E
lie within the circle, on the circle, or outside the circle K.

Solution:

Distance from origin:

• A = √65
• B = √65
• C = √65

Same → lie on a circle

Radius = √65

For D, E:

• D < √65 → inside
• E > √65 → outside

Final Answer:

(i) Yes; radius = √65
(ii) D inside, E outside

Q13. The midpoints of the sides of triangle ABC are the points D, E, and
F. Given that the coordinates of D, E, and F are (5, 1), (6, 5), and
(0, 3), respectively, find the coordinates of A, B and C.

Solution:

Using midpoint reversal:

A = (–1, –1)
B = (11, 3)
C = (1, 7)

Final Answer:

A = (–1, –1), B = (11, 3), C = (1, 7)

Q14. A city has two main roads which cross each other at the centre
of the city. These two roads are along the North–South (N–S)
direction and East–West (E–W) direction. All the other streets of
the city run parallel to these roads and are 200 m apart. There are
10 streets in each direction.
(i) Using 1 cm = 200 m, draw a model of the city in your notebook.
Represent the roads/streets by single lines.
(ii) There are street intersections in the model. Each street An An
intersection is formed by two streets — one running in the
N–S direction and another in the E–W direction.

Each street intersection is referred to in the following manner:

If the Second Street runs in the N–S direction and the 5th Street in
The E–W direction meets at some crossing, then we call this
street intersection (2, 5). Using this convention, find:
(a) How many street intersections can be referred to as (4, 3)?
(b) How many street intersections can be referred to as (3, 4)?

Solution:

Q15. A computer graphics program displays images on a rectangular
screen whose coordinate system has the origin at the
bottom-left corner. The screen is 800 pixels wide and 600 pixels
high. A circular icon of radius 80 pixels is drawn with its centre at
the point A (100, 150). Another circular icon of radius 100 pixels is
drawn with its centre at the point B (250, 230).

Determine:

(i) whether any part of either circle lies outside the screen.
(ii) whether the two circles intersect each other.

Solution:

Q16. Plot the points A (2, 1), B (–1, 2), C (–2, –1), and D (1, –2) in the
coordinate plane. Is ABCD a square? Can you explain why? What
is the area of this square?

Solution: