Orienting Yourself: The Use of Coordinates

In which quadrant does the point \((-4, 7)\) lie?

1. aQuadrant I
2. bQuadrant III
3. cQuadrant II
4. dQuadrant IV

A point lies on the \(y\)-axis, 6 units below the origin. What are its coordinates?

1. a\((6, 0)\)
2. b\((-6, 0)\)
3. c\((0, 6)\)
4. d\((0, -6)\)

What is the distance between the points P \((-2, 3)\) and Q \((4, 11)\)?

1. a\(\sqrt{68}\) units
2. b14 units
3. c10 units
4. d8 units

According to the chapter, the four-quadrant Cartesian plane would be impossible without Brahmagupta’s work. What did Brahmagupta do?

1. aHe formalised zero and negative numbers as algebraic entities
2. bHe fixed Ujjayinī as the central longitude meridian
3. cHe laid out city streets on a precise grid
4. dHe described the latitudes and longitudes of thousands of places

Assertion (A): The point \((-3.5, 0)\) lies on the \(x\)-axis, to the left of the origin.

Reason (R): A point \((x, 0)\) lies on the \(x\)-axis, and if \(x\) is negative, it lies to the left of O.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. Every point in Quadrant III has both coordinates negative.
2. II. The point \((5, -2)\) lies in Quadrant II.
3. III. The \(x\)-coordinate of every point on the \(y\)-axis is 0.

Which of the statements above about quadrants and axes is/are correct?

1. aOnly I
2. bOnly I and III
3. cOnly II and III
4. dI, II and III

Match each point in List I with its position in List II:

1. a(i)-Q, (ii)-S, (iii)-P, (iv)-R
2. b(i)-R, (ii)-S, (iii)-P, (iv)-Q
3. c(i)-S, (ii)-R, (iii)-P, (iv)-Q
4. d(i)-R, (ii)-P, (iii)-S, (iv)-Q

On the floor plan of Reiaan’s room (1 unit = 1 foot), Shalini places a rectangular rug with three of its corners at \((2, 3)\), \((8, 3)\) and \((8, 7)\). Where is the fourth corner, and what is the area of the rug?

1. a\((7, 2)\); 24 square feet
2. b\((2, 7)\); 20 square feet
3. c\((2, 7)\); 24 square feet
4. d\((8, 2)\); 48 square feet

Triangle ABC has vertices A \((2, 5)\), B \((6, 2)\) and C \((2, 2)\). It is reflected in the \(y\)-axis to give triangle \(A'B'C'\). Which statement is correct?

1. a\(A' = (-2, -5)\) and \(A'B' = 5\) units
2. b\(A' = (2, -5)\) and \(A'B' = 5\) units
3. c\(A' = (-2, 5)\) and \(A'B' = 7\) units
4. d\(A' = (-2, 5)\) and \(A'B' = 5\) units

Assertion (A): The points \((3, 8)\) and \((8, 3)\) are the same point of the plane.

Reason (R): If \(x \ne y\), then \((x, y) \ne (y, x)\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. The distance between \((-4, 6)\) and \((5, 6)\) is 1 unit.
2. II. The distance between \((2, -3)\) and \((2, 9)\) is 12 units.
3. III. The distance between \((0, 0)\) and \((-5, -12)\) is 13 units.

Which of the statements above about distances is/are correct?

1. aOnly II and III
2. bOnly II
3. cOnly I
4. dI, II and III

Point W has \(x\)-coordinate \(-5\). Point H lies on the line through W parallel to the \(y\)-axis. Which of these could be H?

1. a\((5, -3)\), in Quadrant IV
2. b\((-5, 3)\), in Quadrant I
3. c\((-3, -5)\), in Quadrant III
4. d\((-5, -3)\), in Quadrant III

A circle K has its centre at the origin O \((0, 0)\) and passes through A \((6, 8)\). Which of these points also lies on K?

1. a\((8, -5)\)
2. b\((-8, -6)\)
3. c\((10, 10)\)
4. d\((-7, 7)\)

Match each pair of points in List I with the distance between them in List II:

1. a(i)-P, (ii)-R, (iii)-S, (iv)-Q
2. b(i)-Q, (ii)-S, (iii)-R, (iv)-P
3. c(i)-Q, (ii)-R, (iii)-S, (iv)-P
4. d(i)-S, (ii)-R, (iii)-Q, (iv)-P

A game screen is 800 pixels wide and 600 pixels high, with the origin at its bottom-left corner. Riya wants to draw a circular icon of radius 50 pixels that stays fully on the screen. Which centre should she choose?

1. a\((700, 100)\)
2. b\((400, 560)\)
3. c\((40, 300)\)
4. d\((780, 300)\)

For a point P with coordinates \((x, y)\), what does \(x\) represent?

1. aDistance of P from the \(x\)-axis, measured along the \(y\)-axis
2. bDistance of P from the \(y\)-axis, measured along the \(x\)-axis
3. cStraight-line distance of P from the origin O
4. dDistance of P from the origin, measured along the \(y\)-axis

Assertion (A): The distance between the points \((2, 1)\) and \((10, 16)\) is 17 units.

Reason (R): The distance between the points \((x_{1}, y)\) and \((x_{2}, y)\) is \(|x_{2} - x_{1}|\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

The points A \((3, 1)\), B \((-1, 3)\), C \((-3, -1)\) and D \((1, -3)\) are the vertices of a square ABCD. What is its area?

1. a\(\sqrt{20}\) square units
2. b40 square units
3. c20 square units
4. d16 square units

Tara uses distances to check the points P \((1, 2)\), Q \((4, 6)\) and R \((10, 14)\). Which conclusion is correct?

1. aThey lie on one straight line, because \(PQ + QR = PR\)
2. bThey form a right-angled triangle, because \(PQ^{2} + QR^{2} = PR^{2}\)
3. cThey are not on one line, because \(PQ + QR \ne PR\)
4. dThey form an isosceles triangle, because \(PQ = QR\)

1. I. Āryabhaṭa replaced the Greek ‘chords’ with ‘sines’, making it easier to calculate the coordinates of a star or a city.
2. II. Al-Bīrūnī perfected the astrolabe, a handheld device that let sailors find their coordinates by looking at the stars.

Which of the statements above is/are correct?

1. aNeither I nor II
2. bOnly I
3. cOnly II
4. dBoth I and II