Orienting Yourself: The Use of Coordinates

20 Questions | 70 Marks

Reiaan's mother has a rectangular garden plot. She wants to plant roses in one corner and wants to mark the boundaries using stakes. She places the first stake at the origin (0, 0). The second stake is placed 10 meters east along the x-axis, and the third stake is placed 15 meters north along the y-axis.

(i) What are the coordinates of the second and third stakes? [1 mark]
(ii) If the fourth stake is placed such that it completes a rectangle with the other three, what are its coordinates? [1 mark]
(iii) Reiaan wants to place a birdbath exactly in the middle of the garden plot. What would be the coordinates of the birdbath? [2 marks]
(iv) The mother decides to plant a small tree at a point which is 5 meters east and 8 meters north of the origin. What are the coordinates of the tree? [1 mark]

A school is planning a sports day event. They have marked a rectangular field for the track events. The corners of the field are at A(– 5, 3), B(5, 3), C(5, – 3), and D(– 5, – 3).

(i) What is the length of the side AB? [1 mark]
(ii) What is the length of the side BC? [1 mark]
(iii) What is the shape of the sports field? Justify your answer. [2 marks]
(iv) A starting point for a race is marked at (0, 0). Is this point inside the rectangular field? [1 mark]

A map of a city shows several landmarks. The library is located at L(– 4, 2), the park is at P(3, 5), and the museum is at M(– 3, – 4).

(i) Which quadrant does the library lie in? [1 mark]
(ii) Which quadrant does the park lie in? [1 mark]
(iii) Which quadrant does the museum lie in? [1 mark]
(iv) If a new cafe is opened at C(3, – 4), which landmark is it closest to, the park or the museum? Justify your answer using the distance concept. [2 marks]

Assertion (A): The point (– 3, 5) lies in the second quadrant.Reason (R): In the second quadrant, the x-coordinate is negative and the y-coordinate is positive.(a) Both A and R are true and R is the correct explanation of A.(b) Both A and R are true but R is not the correct explanation of A.(c) A is true but R is false.(d) A is false but R is true. 1mark

Assertion (A): The distance between points (2, 3) and (2, 8) is 5 units.Reason (R): The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula

\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}

.(a) Both A and R are true and R is the correct explanation of A.(b) Both A and R are true but R is not the correct explanation of A.(c) A is true but R is false.(d) A is false but R is true. 1mark

A ship is located at coordinates (3, 4) on a map. It needs to reach a port located at (7, 1). Calculate the distance the ship needs to travel. 2marks

Two friends, Priya and Ravi, are playing a game using a coordinate grid. Priya marks a point A at (– 2, – 3). Ravi marks a point B such that it is the reflection of A across the x-axis. What are the coordinates of point B? 2marks

A family is planning a road trip. They start from their home at point O (0, 0). They first drive 20 km east to reach point A, and then 30 km north to reach point B.(i) What are the coordinates of point A? [1 mark](ii) What are the coordinates of point B? [1 mark](iii) Calculate the straight-line distance between their home and their final destination B. 2marks

A cartographer is drawing a map of a new area. He places the origin (0,0) at the base of a mountain. A river flows along the positive y-axis. A road starts from point A (5, 0) and goes to point B (5, 12).

(i) What is the length of the road? [1 mark]
(ii) If a bridge is to be built at the midpoint of the road AB, what will be the coordinates of the bridge? [2 marks]

Consider a point P (– 6, 8).(i) What is the distance of P from the y-axis? [1 mark](ii) What is the distance of P from the x-axis? [1 mark](iii) In which quadrant does P lie? 1mark

A company is designing a new logo. The logo has a central point at the origin (0, 0). Two other points, X and Y, are part of the logo. Point X is at (4, 3) and point Y is at (– 4, 3).(i) What is the distance of point X from the origin? [2 marks](ii) What is the distance of point Y from the origin? [2 marks](iii) Are points X and Y equidistant from the origin? Justify. 1mark

A surveyor is mapping a plot of land. He identifies four corners of a rectangular plot as P(1, 1), Q(7, 1), R(7, 5), and S(1, 5).(i) Calculate the length of the diagonal PR. [2 marks](ii) Calculate the length of the diagonal QS. [2 marks](iii) Do the diagonals of this rectangular plot have equal lengths? 1mark

A satellite is orbiting Earth. Its position at a certain time is given by the coordinates (– 8, – 15).(i) What is the distance of the satellite from the origin? [2 marks](ii) If the satellite moves to a new position (– 8, – 10), has it moved closer to or farther from the origin? 2marks

A student is plotting points on a graph paper to represent a geometric shape. They plot the points A(– 3, 4), B(0, 6), C(3, 4), and D(0, 2).(i) What is the shape formed by joining these points in order? [2 marks](ii) Calculate the distance between point A and point D. [2 marks](iii) Calculate the distance between point B and point D. 1mark

A teacher asks a student to plot the points A(– 4, 0), B(– 1, 0), and C(2, 0).(i) Are these points on the same line? [1 mark](ii) Which axis do these points lie on? [1 mark](iii) What can be said about the y-coordinates of points lying on the x-axis? 1mark

A point P is located at (5, – 2). A point Q is located at (– 1, – 2).(i) What is the distance between P and Q? [2 marks](ii) What does the fact that both points have the same y-coordinate tell you about the line segment PQ? 1mark

A computer game character is at position (x, y). The character moves 3 units to the left and 4 units up. If the new position is (2, 7), what was the original position (x, y) of the character? 3marks

A surveyor is marking points for a construction project. She marks a point P at (6, 8). She then marks a point Q such that it is the reflection of P across the y-axis. What are the coordinates of Q? 2marks

A circular park has its center at the origin (0, 0). A bench is located at point B (6, 8).(i) Calculate the distance of the bench from the center of the park. [2 marks](ii) If the radius of the park is 10 units, is the bench located on the boundary of the park? 1mark

A researcher is studying the movement of a particle. The particle starts at (– 3, 5). It then moves to (– 3, – 7).(i) What is the distance the particle moved? [2 marks](ii) Which axis is the particle moving parallel to? [1 mark](iii) What can be said about the x-coordinates of points lying on a line parallel to the y-axis? 1mark

Answers

(i) Second stake: (10, 0). Third stake: (0, 15).(ii) The fourth stake will be at (10, 15).(iii) The midpoint of the diagonal connecting (0, 0) and (10, 15) is

(\frac{0 + 10}{2}, \frac{0 + 15}{2}) = (5, 7.5)

.(iv) The coordinates of the tree are (5, 8).

(i) Distance AB =

\sqrt{(5 - (- 5))^{2} + (3 - 3)^{2}} = \sqrt{10^{2} + 0^{2}} = 10

units.(ii) Distance BC =

\sqrt{(5 - 5)^{2} + (- 3 - 3)^{2}} = \sqrt{0^{2} + (- 6)^{2}} = 6

units.(iii) The shape is a rectangle. The opposite sides AB and CD have length 10 units, and BC and AD have length 6 units. Also, AB is parallel to the x-axis, and BC is parallel to the y-axis, so adjacent sides are perpendicular.(iv) The point (0, 0) has an x-coordinate between – 5 and 5, and a y-coordinate between – 3 and 3. Therefore, it is inside the rectangular field.

(i) Quadrant II.(ii) Quadrant I.(iii) Quadrant III.(iv) Distance LP =

\sqrt{(3 - (- 4))^{2} + (5 - 2)^{2}} = \sqrt{7^{2} + 3^{2}} = \sqrt{49 + 9} = \sqrt{58}

units.Distance MP =

\sqrt{(3 - (- 3))^{2} + (5 - (- 4))^{2}} = \sqrt{6^{2} + 9^{2}} = \sqrt{36 + 81} = \sqrt{117}

units.Since

\sqrt{58} < \sqrt{117}

, the cafe is closest to the park.

(a) Both A and R are true and R is the correct explanation of A.

(b) Both A and R are true but R is not the correct explanation of A. (While the formula is correct, the question asks for a specific distance. The formula is not strictly needed to determine the distance between (2, 3) and (2, 8) as the x-coordinates are the same.)

Distance =

\sqrt{(7 - 3)^{2} + (1 - 4)^{2}} = \sqrt{4^{2} + (- 3)^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5

units.

When a point (x, y) is reflected across the x-axis, its image is (x, -y). So, the reflection of A(– 2, – 3) across the x-axis is B(– 2, 3).

(i) Point A: (20, 0).(ii) Point B: (20, 30).(iii) Distance OB =

\sqrt{(20 - 0)^{2} + (30 - 0)^{2}} = \sqrt{20^{2} + 30^{2}} = \sqrt{400 + 900} = \sqrt{1300} = 10 \sqrt{13}

km.

(i) The road is a vertical line segment. Its length is the difference in the y-coordinates: 12 - 0 = 12 units.(ii) Midpoint =

(\frac{5 + 5}{2}, \frac{0 + 12}{2}) = (5, 6)

(i) The distance of P from the y-axis is its x-coordinate's absolute value, which is

∣ - 6 ∣ = 6

units.(ii) The distance of P from the x-axis is its y-coordinate's absolute value, which is

∣ 8 ∣ = 8

units.(iii) Quadrant II.

(i) Distance OX =

\sqrt{(4 - 0)^{2} + (3 - 0)^{2}} = \sqrt{4^{2} + 3^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5

units.(ii) Distance OY =

\sqrt{(- 4 - 0)^{2} + (3 - 0)^{2}} = \sqrt{(- 4)^{2} + 3^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5

units.(iii) Yes, points X and Y are equidistant from the origin because their distances calculated above are equal.

(i) Diagonal PR =

\sqrt{(7 - 1)^{2} + (5 - 1)^{2}} = \sqrt{6^{2} + 4^{2}} = \sqrt{36 + 16} = \sqrt{52}

units.(ii) Diagonal QS =

\sqrt{(1 - 7)^{2} + (5 - 1)^{2}} = \sqrt{(- 6)^{2} + 4^{2}} = \sqrt{36 + 16} = \sqrt{52}

units.(iii) Yes, the diagonals of this rectangular plot have equal lengths.

(i) Distance from origin =

\sqrt{(- 8)^{2} + (- 15)^{2}} = \sqrt{64 + 225} = \sqrt{289} = 17

units.(ii) New distance from origin =

\sqrt{(- 8)^{2} + (- 10)^{2}} = \sqrt{64 + 100} = \sqrt{164}

units.Since

\sqrt{164} < 17

, the satellite has moved closer to the origin.

(i) The shape formed is a kite. Sides AB and BC have length

\sqrt{3^{2} + 2^{2}} = \sqrt{13}

. Sides CD and DA have length

\sqrt{(- 3)^{2} + (- 2)^{2}} = \sqrt{13}

. AB = BC = CD = DA =

\sqrt{13}

is not true.AB =

\sqrt{(0 - (- 3))^{2} + (6 - 4)^{2}} = \sqrt{3^{2} + 2^{2}} = \sqrt{9 + 4} = \sqrt{13}

BC =

\sqrt{(3 - 0)^{2} + (4 - 6)^{2}} = \sqrt{3^{2} + (- 2)^{2}} = \sqrt{9 + 4} = \sqrt{13}

CD =

\sqrt{(0 - 3)^{2} + (2 - 4)^{2}} = \sqrt{(- 3)^{2} + (- 2)^{2}} = \sqrt{9 + 4} = \sqrt{13}

DA =

\sqrt{(- 3 - 0)^{2} + (4 - 2)^{2}} = \sqrt{(- 3)^{2} + 2^{2}} = \sqrt{9 + 4} = \sqrt{13}

All sides are equal, so it is a rhombus. Since the diagonals are on the axes and are not equal (BD length = 8, AC length = 6), it's a rhombus. However, visually plotting suggests a kite. Let's recheck the question's intent.The points are A(– 3, 4), B(0, 6), C(3, 4), and D(0, 2).AB =

\sqrt{13}

, BC =

\sqrt{13}

, CD =

\sqrt{13}

, DA =

\sqrt{13}

. It's a rhombus.(ii) Distance AD =

\sqrt{(0 - (- 3))^{2} + (2 - 4)^{2}} = \sqrt{3^{2} + (- 2)^{2}} = \sqrt{9 + 4} = \sqrt{13}

units.(iii) Distance BD =

\sqrt{(0 - 0)^{2} + (2 - 6)^{2}} = \sqrt{0^{2} + (- 4)^{2}} = \sqrt{16} = 4

units.

*Correction for 14(i) based on calculated distances: The shape is a rhombus.

(i) Yes, the points A, B, and C are on the same line because they all have a y-coordinate of 0.(ii) These points lie on the x-axis.(iii) The y-coordinates of points lying on the x-axis are always 0.

(i) The y-coordinates are the same, so the distance is the absolute difference of the x-coordinates:

∣ 5 - (- 1) ∣ = ∣ 5 + 1 ∣ = 6

units.(ii) The line segment PQ is parallel to the x-axis.

Let the original position be (x, y).Moving 3 units left means the new x-coordinate is x - 3.Moving 4 units up means the new y-coordinate is y + 4.The new position is (x - 3, y + 4), which is given as (2, 7).So, x - 3 = 2 => x = 5.And y + 4 = 7 => y = 3.The original position was (5, 3).

When a point (x, y) is reflected across the y-axis, its image is (– x, y).So, the reflection of P(6, 8) across the y-axis is Q(– 6, 8).

(i) Distance OB =

\sqrt{(6 - 0)^{2} + (8 - 0)^{2}} = \sqrt{6^{2} + 8^{2}} = \sqrt{36 + 64} = \sqrt{100} = 10

units.(ii) Yes, the bench is located on the boundary of the park because its distance from the center (10 units) is equal to the radius of the park.

(i) The x-coordinates are the same, so the distance moved is the absolute difference of the y-coordinates:

∣ - 7 - 5 ∣ = ∣ - 12 ∣ = 12

units.(ii) The particle is moving parallel to the y-axis.(iii) The x-coordinates of points lying on a line parallel to the y-axis are constant.

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Orienting Yourself: The Use of Coordinates — Competency Based Questions

Competency based questions

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