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Orienting Yourself : The Use of Coordinates

CBSE · Class 9 · Mathematics

NCERT Solutions for Orienting Yourself : The Use of Coordinates — CBSE Class 9 Mathematics.

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54 Questions Solved · 4 Sections

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EXERCISE SET 1.1

1(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?Show solution
From the figure, the door segment D1R1D_1R_1 lies on the x-axis.
- So its distance from the y-axis is the **x-coordinate of D1D_1.
- Its distance from the x-axis is 0**, because it is on the x-axis.

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1(ii)What are the coordinates of D₁?Show solution
Since the door lies on the x-axis and the room’s left wall is the y-axis, the left end of the door is at x=8x=8 in the figure. Hence the coordinates of D1D_1 are **(8,0)(8,0)**.

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1(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?Show solution
The width of the door is the distance from D1D_1 to R1R_1:
11.5−8=3.5 ft 11.5-8=3.5\text{ ft}
So the door is 3.5 ft wide. This is a comfortable width for a room door. A wheelchair user would be able to enter easily, since the width is adequate.

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1(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?Show solution
The bathroom door runs from B1(0,1.5)B_1(0,1.5) to B2(0,4)B_2(0,4), so its width is
4−1.5=2.5 ft. 4-1.5=2.5\text{ ft}.
The room door is 11.5−8=3.511.5-8=3.5 ft wide. Since 2.5<3.52.5<3.5, the bathroom door is narrower than the room door.

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2What are the standard widths for a room door? Look around your home and in school.Show solution

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3Are the doors in your school suitable for people in wheelchairs?Show solution

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Think and Reflect

1What is the x-coordinate of a point on the y-axis?Show solution
A point on the y-axis has x-coordinate 0.

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2Is there a similar generalisation for a point on the x-axis?Show solution
Yes. Similarly, a point on the x-axis has y-coordinate 0.

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3Does point Q (y, x) ever coincide with point P (x, y)? Justify your answer.Show solution
A point Q(y,x)Q(y,x) coincides with P(x,y)P(x,y) only when the two coordinates are equal, i.e. when x=yx=y. If x≠yx\neq y, then the ordered pairs are different, so the points do not coincide.

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4If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y. Is this claim true?Show solution
Yes, the claim is true.
- If x≠yx\neq y, then the ordered pairs (x,y)(x,y) and (y,x)(y,x) are different.
- If (x,y)=(y,x)(x,y)=(y,x), then the first coordinates and second coordinates must be equal, so x=yx=y.
Thus, (x,y)=(y,x)(x,y)=(y,x) if and only if x=yx=y.

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1In 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?Show solution
From A(3,4)A(3,4) to D(7,1)D(7,1):
- Along the x-axis: 7−3=47-3=4 units
- Along the y-axis: 4−1=34-1=3 units

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2Can these distances help you find the distance AD?Show solution
Yes. These distances form the two perpendicular sides of a right triangle, so by the Baudhāyana–Pythagoras theorem the distance ADAD is
AD=42+32=16+9=25=5 units. AD=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5\text{ units.}

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1What has remained the same and what has changed with this reflection?Show solution
With reflection in the y-axis, the shape and lengths remain the same, but the x-coordinates change sign. So what remains unchanged is the size and side lengths of the triangle; what changes is its position/orientation on the plane.

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2Would these observations be the same if Δ\DeltaADM is reflected in the x-axis (instead of the y-axis)?Show solution
Yes. Reflection in the x-axis would also preserve the lengths and shape of the triangle. In that case, the y-coordinates would change sign while the x-coordinates stay the same.

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EXERCISE SET 1.2

1Place Reiaan's rectangular study table with three of its feet at the points (8, 9), (11, 9) and (11, 7).Show solution

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1(i)Where will the fourth foot of the table be?Show solution
Three feet are at (8,9)(8,9), (11,9)(11,9) and (11,7)(11,7). The missing fourth corner of the rectangle must have x-coordinate 88 and y-coordinate 77.
So the fourth foot is at **(8,7)(8,7)**.

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1(ii)Is this a good spot for the table?Show solution
Yes, it is a good spot if the table fits without blocking movement. Since the feet form a rectangle in the given space, the table is placed neatly and properly in the room.

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1(iii)What is the width of the table? The length? Can you make out the height of the table?Show solution
From the coordinates, the horizontal side is 11−8=311-8=3 ft and the vertical side is 9−7=29-7=2 ft. So the table is 3 ft by 2 ft. Since this is a floor plan, the height cannot be determined from the coordinates.

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2If 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?Show solution
The bathroom door opens from hinge B1B_1. Whether it hits the wardrobe depends on the exact positions shown in the figure. From the given layout, it does not hit the wardrobe if opened as shown. If the door is made wider, it may need more clearance, so shifting the wardrobe or changing the door swing could be suggested.

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3Look at Reiaan's bathroom.Show solution

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3(i)What are the coordinates of the four corners O, F, R, and P of the bathroom?Show solution
From the bathroom figure, the corners are read directly as:
- O at (0,0)(0,0)
- F at (0,5)(0,5)
- R at (8,5)(8,5)
- P at (8,0)(8,0)

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3(ii)What is the shape of the showering area SHWR in Reiaan's bathroom? Write the coordinates of the four corners.Show solution
The showering area SHWR is a rectangle. Its four corners are:
- S (2,2)(2,2)
- H (2,5)(2,5)
- W (5,5)(5,5)
- R (5,2)(5,2)

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3(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.Show solution

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4Other rooms in the house:Show solution

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4(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.Show solution
The dining room is a rectangle of length 18 ft and width 15 ft, with the length along PAPA. So, if PP is at the origin of that room layout, then the opposite corner AA will be 1818 ft along the length and 1515 ft along the width. The four corners can be marked accordingly on the sketch as a rectangle.

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4(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.Show solution
A 5 ft×3 ft5\text{ ft} \times 3\text{ ft} table placed at the centre of an 18 ft×15 ft18\text{ ft} \times 15\text{ ft} room will have its feet at positions equally spaced around the centre of the room. The exact coordinates depend on the coordinate choice in the sketch, so the answer is to mark a centred rectangle and write the feet accordingly.

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END-OF-CHAPTER EXERCISES

1What are the x-coordinate and y-coordinate of the point of intersection of the two axes?Show solution
The point where the x-axis and y-axis intersect is the origin, whose coordinates are **(0,0)(0,0)**.

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2Point 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?
3Consider the points R (3, 0), A (0, -2), M (-5, -2) and P (-5, 2). If they are joined in the same order, predict:
3(i)Two sides of RAMP that are perpendicular to each other.
3(ii)One side of RAMP that is parallel to one of the axes.
3(iii)Two points that are mirror images of each other in one axis. Which axis will this be?
4Plot point Z (5, -6) on the Cartesian plane. Construct a right-angled triangle IZN and find the lengths of the three sides.
5What 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?
6Are 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.
7Use 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.
8Using the origin as one vertex, plot the vertices of:
*8(i)A right-angled isosceles triangle.
*8(ii)An isosceles triangle with one vertex in Quadrant III and the other in Quadrant IV.
9The following table shows the coordinates of points S, M and T. In each case, state whether M is the midpoint of segment ST. Justify your answer.
10Use 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).
11Let 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).
12(i)Given the points A (1, -8), B (-4, 7) and C (-7, -4), show that they lie on a circle K whose center is the origin O (0, 0). What is the radius of circle K?
12(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.
13The 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.
14A 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.
14(i)Using 1 cm = 200 m, draw a model of the city in your notebook. Represent the roads/streets by single lines.
14(ii)There are street intersections in the model. Each street intersection is formed by two streets—one running in the N–S direction and another in the E–W direction. Each street
14(ii)(a)how many street intersections can be referred to as (4, 3).
14(ii)(b)how many street intersections can be referred to as (3, 4).
15A 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:
15(i)whether any part of either circle lies outside the screen.
15(ii)whether the two circles intersect each other.
16Plot 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?

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Frequently Asked Questions

What are the important topics in Orienting Yourself : The Use of Coordinates for CBSE Class 9 Mathematics?
Orienting Yourself : The Use of Coordinates covers several key topics that are frequently asked in CBSE Class 9 board exams. Focus on the core concepts listed on this page and practise related questions to build confidence.
How to score full marks in Orienting Yourself : The Use of Coordinates — CBSE Class 9 Mathematics?
Understand the core concepts first, then work through the 89 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.
Where can I get free NCERT Solutions for Orienting Yourself : The Use of Coordinates Class 9 Mathematics?
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