1
The Sindhu-Sarasvatī Civilisation used a coordinate system for urban planning. In what directions were their city streets constructed? 1mark
2
What was the approximate distance between the North-South and East-West streets in the Sindhu-Sarasvatī Civilisation? 1mark
3
Who is credited with replacing Greek 'chords' with 'sines' for easier calculation of celestial coordinates? 1mark
4
What fundamental concept, formalized by Brahmagupta, is essential for the four-quadrant Cartesian plane? 1mark
5
Al-Bīrūnī used a device to help sailors find their coordinates by looking at the stars. What was this device called? 1mark
6
René Descartes, following Fermat's work, formalized the idea that any point in a two-dimensional plane can be defined by how many numbers? 1mark
7
Shalini uses a scale of 1 cm : 1 foot to create a sketch of Reiaan's room. If a pin is placed at a point representing a corner of the room, and its coordinate is (5, 3), what does this mean in terms of feet from the origin and axes? 2marks
8
In the Cartesian coordinate system, what are the two lines called that are at right angles to each other, and what is the point of their intersection called? 2marks
9
A point P has coordinates (– 7, 5).(i) In which quadrant does point P lie?(ii) What is the perpendicular distance of P from the x-axis? 2marks
10
A point Q has coordinates (4, – 3).(i) In which quadrant does point Q lie?(ii) What is the perpendicular distance of Q from the y-axis? 2marks
11
Consider the points A (– 3, 5) and B (5, 5).(i) What is the distance between points A and B?(ii) Which axis is the line segment AB parallel to? 3marks
12
Consider the points C (2, – 4) and D (2, 6).(i) What is the distance between points C and D?(ii) Which axis is the line segment CD parallel to? 3marks
13
Plot the points P (– 2, 3), Q (4, 5), R (6, – 2), and S (– 4, – 4) on a graph sheet.(i) Identify the quadrant in which each point lies.(ii) Determine which of these points lie on the axes. 4marks
14
The points A (1, 2), B (4, 2) and C (4, 5) are three vertices of a rectangle ABCD. Plot these points and find the coordinates of the fourth vertex D. Justify your answer. 3marks
15
A rectangular study table has three of its feet at the points (7, 8), (10, 8) and (10, 6).(i) What are the coordinates of the fourth foot of the table?(ii) What is the width of the table? 3marks
16
Reiaan’s room has corners marked at O (0, 0), A (12, 0), B (12, 10) and C (0, 10).(i) If the door D is located at (9, 0), how far is the door from the left wall (y-axis)?(ii) How far is the door from the x-axis? 3marks
17
Calculate the distance between the points (– 3, 4) and (5, – 2). 3marks
18
Reiaan wants to mark a 3 ft × 2 ft space for a washbasin in his room. The available space has corners at (0, 6) and (3, 6).(i) What could be the coordinates of the other two corners of the washbasin area?(ii) If the washbasin is placed with one corner at (0, 6) and it extends 3 ft along the x-axis and 2 ft along the y-axis, what are the coordinates of its four corners? 4marks
19
A programmer is designing a video game. A character is located at point P (– 2, – 3). The character moves 5 units to the right and 7 units up.(i) What are the new coordinates of the character?(ii) In which quadrant is the character after the move? 3marks
20
The table shows the coordinates of points S, M and T. In each case, state whether M is the midpoint of segment ST and justify your answer.S M T(– 3, 0) (0, 0) (3, 0)(2, 3) (3, 4) (4, 5)(0, 0) (0, 5) (0, – 10) 5marks
Answers
1
The city streets were constructed in North-South and East-West directions.
2
The distance between the streets was about 10 metres.
3
Āryabhaṭa is credited with replacing Greek 'chords' with 'sines'.
4
The concept of zero and negative numbers is essential for the four-quadrant Cartesian plane.
5
The device was called an 'astrolabe'.
6
Two numbers are needed to define any point in a two-dimensional plane.
7
(i) The point (5, 3) means it is 5 feet to the right of the origin along the x-axis and 3 feet up from the origin along the y-axis.(ii) The perpendicular distance from the origin along the x-axis is 5 feet, and the perpendicular distance from the origin along the y-axis is 3 feet.
8
The two lines are called the x-axis and the y-axis. The point of their intersection is called the origin.
9
(i) Point P (– 7, 5) lies in Quadrant II (negative x-coordinate, positive y-coordinate).(ii) The perpendicular distance of P from the x-axis is the absolute value of its y-coordinate, which is units.
10
(i) Point Q (4, – 3) lies in Quadrant IV (positive x-coordinate, negative y-coordinate).(ii) The perpendicular distance of Q from the y-axis is the absolute value of its x-coordinate, which is units.
11
(i) Since both points have the same y-coordinate (5), they lie on a horizontal line. The distance is the absolute difference of their x-coordinates: units.(ii) The line segment AB is parallel to the x-axis.
12
(i) Since both points have the same x-coordinate (2), they lie on a vertical line. The distance is the absolute difference of their y-coordinates: units.(ii) The line segment CD is parallel to the y-axis.
13
(i) P (– 2, 3) lies in Quadrant II.Q (4, 5) lies in Quadrant I.R (6, – 2) lies in Quadrant IV.S (– 4, – 4) lies in Quadrant III.(ii) None of these points lie on the axes, as neither the x-coordinate nor the y-coordinate is zero for any of them.
14
Plotting A (1, 2), B (4, 2), and C (4, 5).AB is horizontal (y-coordinates are the same), length = .BC is vertical (x-coordinates are the same), length = .Since AB is parallel to the x-axis and BC is parallel to the y-axis, they are perpendicular.For ABCD to be a rectangle, AD must be parallel to BC and have the same length, and CD must be parallel to AB and have the same length.So, D must have the same x-coordinate as A (1) and the same y-coordinate as C (5).Therefore, the coordinates of D are (1, 5).
15
(i) Let the points be A (7, 8), B (10, 8), C (10, 6). Since ABCD is a rectangle, the fourth foot D will have the x-coordinate of A (7) and the y-coordinate of C (6). So, D is (7, 6).(ii) Width (parallel to x-axis): ft. Length (parallel to y-axis): ft. The width of the table is 3 ft.
16
(i) The left wall is the y-axis. The distance of point D (9, 0) from the y-axis is its x-coordinate, which is 9 ft.(ii) The distance of point D (9, 0) from the x-axis is its y-coordinate, which is 0 ft. (The door is on the x-axis, so it's on the floor, no distance from the floor).
17
Let the points be and .Distance
units.
18
(i) Let the corners be P (0, 6) and Q (3, 6). Since it's a 3 ft x 2 ft space, and PQ is along the x-axis direction (length 3 ft), the other dimension must be 2 ft along the y-axis.The other two corners could be R (3, 6 - 2) = (3, 4) and S (0, 6 - 2) = (0, 4).(ii) The four corners are (0, 6), (3, 6), (3, 4), and (0, 4).
19
(i) The character starts at P (– 2, – 3).Moves 5 units to the right: x-coordinate becomes .Moves 7 units up: y-coordinate becomes .The new coordinates are (3, 4).(ii) The new coordinates (3, 4) have a positive x-coordinate and a positive y-coordinate, so the character is in Quadrant I.
20
S M T Is M the midpoint Reason for yourof ST? Yes or No answer(– 3, 0) (0, 0) (3, 0) Yes The x-coordinate of M () is the average of the x-coordinates of S () and T (): . The y-coordinate of M () is the average of the y-coordinates of S () and T (): .(2, 3) (3, 4) (4, 5) Yes The x-coordinate of M () is the average of the x-coordinates of S () and T (): . The y-coordinate of M () is the average of the y-coordinates of S () and T (): .(0, 0) (0, 5) (0, – 10) No The x-coordinate of M () is the average of the x-coordinates of S () and T (): . However, the y-coordinate of M () is not the average of the y-coordinates of S () and T (): .Connection: When M is the midpoint of ST, the coordinates of M are the average of the corresponding coordinates of S and T. i.e., .
