Coordinate Geometry — Question Bank | Grade 10 CBSE Mathematics

Q1 1 mark MCQ p. 55

What is the distance between the points (−1, 3) and (2, −5)?

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Correct option: 4 — √73

Q2 1 mark MCQ p. 55

A circle of radius 5 units has its centre at (−2, 2). The point (−6, y) lies on the circle.Which of these could be the value of y?

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Correct option: 3 — 5

Q3 1 mark MCQ p. 55

P(1, 7), Q(−3, 2) and R(6, 1) are the coordinates of the vertices of a triangle.Which of the following types of triangle is ΔPQR?

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Correct option: 1 — Scalene triangle

Q4 1 mark MCQ p. 55

In the SQUARE given below, the coordinates of two adjacent vertices P and Q are given.[Figure: square PQRS with P(4, 2) at the top-left corner and S at the top-right corner; Q(4, −2) at the bottom-left corner and R at the bottom-right corner. PQ is the left side of the square.]What are the coordinates of vertex R?

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Correct option: 3 — (8, −2)

Q5 1 mark MCQ p. 55

ΔPQR is a triangle such that PQ:PR = 1:2. Point P lies on the x-axis and the coordinates of Q and R are known.Which of the following formula can DEFINITELY be used to find the coordinates of P?i) Section formulaii) Distance formula

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Correct option: 2 — only ii)

Q6 1 mark MCQ p. 56

Which one of these is the relation between x and y if (x, y) is equidistant from (−1, 4) and (2, 5)?

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Correct option: 3 — 3x + y = 6

Q7 1 mark MCQ p. 56

What is the distance of (7, −3) from the origin?

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Correct option: 4 — √58 units

Q8 1 mark MCQ p. 56

Which of the following points is the mid-point of the line segment joining P(5, 2) and Q(7, 6)?

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Correct option: 2 — (6, 4)

Q9 1 mark Written p. 56

The point (x, y) is equidistant from (−4, 0) and (5, 3).Write an equation relating x and y. Show your steps.

Reveal official answer

• Applies the distance formula correctly to write √{(x + 4)² + y²} = √{(x − 5)² + (y − 3)²} [0.5]

• Writes the relation as 3x + y = 3. [0.5]

Q10 1 mark Written p. 56

In what ratio does the origin divides line segment joining A(−5, 0) and B(3, 0)? Show your work.

Reveal official answer

• Writes that the distance of A from the origin is 5 units and that of B from the origin is 3 units. Hence, the ratio in which the origin divides the line segment AB is 5:3. (Award full marks if student uses any other method using calculation.) [1]

Q11 2 marks Written p. 56

A(6, 8), B(3, 7) and C(4, 4) are the vertices of a right-angled triangle, where ∠B = 90°.Find the area of the triangle. Show your work.

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• Identifies height of triangle = AB and base of triangle = BC. [0.5]

• Finds height = AB = √{(−3)² + (−1)²} = √10 units and base = BC = √{(1)² + (−3)²} = √10 units. [1]

• Finds area of triangle = ½ × base × height = ½ × √10 × √10 = 5 square units. [0.5]

Q12 2 marks Written p. 56

F lies on the line segment joining E(−3, 2) and G(4, 5). F divides EG in the ratio 2:1.Find the coordinates of F. Show your work.

Reveal official answer

• Uses the section formula to find the coordinates of the point F as follows: ((2(4) + 1(−3)) / (2 + 1), (2(5) + 1(2)) / (2 + 1)) [1]

• Simplifies the above expression and finds the coordinates of point F as (5/3, 4). [1]

Q13 2 marks Written p. 57

In the figure given below, AB is the diameter of the circle with centre O and OB is the diameter of the circle with centre C.[Figure: A(1, −2), O(3, 4), C and B lie in that order along the diameter line, with C between O and B.](Note: The figure is not to scale)Find the coordinates of point C. Show your steps.

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• Finds the coordinates of B using the mid-point formula as B(5, 10). Working may look like: Let co-ordinates of B be (x, y). O(3, 4) = ((x + 1)/2, (y − 2)/2) => x = 5, y = 10. [1.5]

• Finds the coordinates of C using the mid-point formula as C(4, 7). Working may look like: Let co-ordinates of C be (m, n). C(m, n) = ((3 + 5)/2, (4 + 10)/2) => m = 4, n = 7. [0.5]

Q14 2 marks Written p. 57

Shown below is a right triangle ABC.[Figure: right angle at B; A(2, 3) and C(−3, −1); AB is vertical (parallel to the y-axis) and CB is horizontal (parallel to the x-axis).]Find the value of cos C. Show your work.

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• Finds the coordinates of B as (2, −1). [0.5]

• Uses the distance formula and finds AC = √(5² + 4²) = √41 units and BC = √(5)² = 5 units. [1]

• Mentions cos C = BC/AC and finds the value as 5/√41. [0.5]

Q15 2 marks Written p. 57

Find the ratio in which O(4, 3) divides the line segment joining A(2, 1) and B(7, 6). Show your work.

Reveal official answer

• Finds the distances using the distance formula: AO = √8 = 2√2 units, BO = √18 = 3√2 units. [1]

• Finds the ratio AO/BO = 2/3. Hence, the ratio in which O(4, 3) divides the line segment AB is 2:3. (Award full marks if the student correctly solves the same using the Section Formula.) [1]

Q16 2 marks Written p. 57

Find the length of the longest side of the triangle formed by the points of intersection of line 8x + 6y = 48 with the coordinate axes. Show your work.

Reveal official answer

• Substitutes x and y as 0 in the given equation 8x + 6y = 48 to find the coordinates of the points of intersection as (0, 8) and (6, 0) respectively. [1]

• Uses the distance formula to find the length of the longest side of the triangle as √{(0 − 6)² + (8 − 0)²} = 10 units. [1]

Q17 2 marks Written p. 58

A square is inscribed in a circle of radius 2 cm with center O at the origin. All 4 vertices of the square lie on the coordinate axes.Use the distance formula to find the length of the side of the square. Show your work.

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• Writes that the coordinates of the vertices of the circle would be (2, 0), (0, −2), (−2, 0), (0, 2). [1]

• Uses the distance formula and any 2 adjacent coordinates of the vertices of the square to find the length of the side of the square as 2√2 cm. [1]

Q18 3 marks Written p. 58

Check whether the points A(0, 5), B(2, 3), C(4, 5) and D(2, 7) are the vertices of a square. Show your work.

Reveal official answer

• Finds the measure of AB as √{(2)² + (−2)²} = √8 = 2√2 units. Finds the measure of BC as √{2² + 2²} = √8 = 2√2 units. [1]

• Finds the measure of CD as √{(−2)² + 2²} = √8 = 2√2 units. Finds the measure of DA as √{(−2)² + (−2)²} = √8 = 2√2 units. [1]

• Finds the diagonals of ABCD as: AC = √{(4 − 0)² + (5 − 5)²} = √16 = 4 units. BD = √{(2 − 2)² + (7 − 3)²} = √16 = 4 units. [0.5]

• Concludes AB = BC = CD = DA and, AC = BD. Hence, A, B, C, and D are vertices of a square. (Award full marks if the student uses any other method to prove this). [0.5]

Q19 3 marks Written p. 58

Atul plotted the seating plan of his classroom on a cartesian plane such that, Abdul is seated at (3, 7), and Vaibhav is seated at (−2, −1). Prashant is seated somewhere on the line that connects Abdul and Vaibhav. It is given that the distance between Prashant and Vaibhav is half of the distance between Abdul and Prashant.What are the coordinates of Prashant's seat? Show your work.

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• Represents the given situation mathematically as: Let the positions of Abdul, Prashant and Vaibhav be as points A, P and V on the seating plan. Here, PV = ½AP => AP/PV = 2/1 => AP:PV = 2:1 [1]

• Uses section formula for the coordinates of P such that it divides AV in the ratio of 2:1 as: ((1(3) + 2(−2)) / (2 + 1), (1(7) + 2(−1)) / (2 + 1)) [1]

• Simplifies the above expression to find the coordinates of Prashant's seat as (−1/3, 5/3). [1]

Q20 3 marks Written p. 58

P(−6, 4) and Q(2, 10) are the two end-points of the diameter of the circle with centre O(x, y).i) Find the radius.ii) Prove that 4x + 3y − 13 = 0.Show your steps.

Reveal official answer

• i) Finds the diameter, PQ as √{(2 + 6)² + (10 − 4)²} = 10 units. [1]

• Finds the radius as 10/2 = 5 units. [0.5]

• ii) Uses the distance formula and writes the following relation: (x + 6)² + (y − 4)² = (x − 2)² + (y − 10)² [0.5]

• Simplifies the above equation and concludes that 4x + 3y − 13 = 0. [1]

Q21 3 marks Written p. 58

Find the ratio in which the x-axis divides the line segment joining the points A(4, 9) and B(3, −5). Show your work.

Reveal official answer

• Assumes that the ratio as p:q and mentions that the coordinates of the point at which the line intersects the x-axis can be taken as (x, 0). [1]

• Uses the section formula to write the equation as: (x, 0) = ((3p + 4q)/(p + q), (−5p + 9q)/(p + q)) [1]

• Equates (−5p + 9q)/(p + q) to 0 as: (−5p + 9q)/(p + q) = 0 => 5p = 9q => p : q = 9:5 [1]

Q22 3 marks Written p. 58

The three vertices of a rhombus ABCD are A(−3, 2), B(−5, −5) and C(2, −3).i) Find the coordinates of the point where both the diagonals AC and BD intersect.ii) Find the coordinates of the fourth vertex D.Show your steps and give valid reasons.

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• i) Writes that the diagonals of a rhombus bisect each other. [0.5]

• Finds the point of intersection of both the diagonals by finding the mid-point of A(−3, 2) and C(2, −3) as (−1/2, −1/2). [0.5]

• ii) Finds the mid-point of B(−5, −5) and D(x, y) as ((x − 5)/2, (y − 5)/2), where x and y are the coordinates of the fourth vertex D. [0.5]

• Uses the above steps and equates the respective coordinates of the mid-points to get the following relationships: −1/2 = (x − 5)/2, −1/2 = (y − 5)/2 [0.5]

• Solves the above two equations to find the values of x and y as 4 and 4 respectively. Concludes that the coordinates of the fourth vertex D are (4, 4). [1]

Q23 3 marks Written p. 58

Prove that A(−1, 1), B(1, 2) and C(3, 3) are collinear.

Reveal official answer

• Assumes that A, B and C are collinear and hence AB + BC = AC. Finds the distance AB, BC and AC as: AB = √(2² + 1²) = √5 units, BC = √(2² + 1²) = √5 units, AC = √(4² + 2²) = √20 = 2√5 units [2]

• Writes that since AB + BC = AC, A, B and C are collinear. (Award full marks if the student proves the same using the area of the triangle method.) [1]

Q24 3 marks Written p. 58

Points C and D divide the line segment AB into 3 equal parts where the coordinates of points A and D are (4,2) and (8,10) respectively.What are the coordinates of point B? Show your work.

Reveal official answer

• Assumes the coordinate of point B as (x, y). States that since points C and D divide line segment AB into 3 equal parts, point D will divide AB in the ratio of 1:2 or 2:1. [1]

• Uses section formula to find the values of (x, y) as (16,26) when D divides AB in ratio 1:2. The working may look as follows: (8,10) = ((x + 8)/3, (y + 4)/3) [1]

• Uses section formula to find the values of (x, y) as (10,14) when D divides AB in ratio 2:1. The working may look as follows: (8,10) = ((2x + 4)/3, (2y + 2)/3) [1]

Q25 5 marks Written p. 59

A circle passes through the following points:P(−1, 5), Q(−4, 6) and R(−2, 2)i) Find the coordinates of the centre of the circle.ii) Find the radius of the circle.Show your work.

Reveal official answer

• i) Assumes the centre of the circle as any point, say O(x, y) and uses the distance formula to find OP, OQ and OR. OP = √[(x + 1)² + (y − 5)²] = √(x² + 2x + y² − 10y + 26), OQ = √[(x + 4)² + (y − 6)²] = √(x² + 8x + y² − 12y + 52), OR = √[(x + 2)² + (y − 2)²] = √(x² + 4x + y² − 4y + 8) [1.5]

• Uses OP = OQ to get 3x − y + 13 = 0. Uses OP = OR to get x + 3y − 9 = 0. Uses OQ = OR to get x − 2y + 11 = 0. (Award full marks if any 2 of the 3 equations are formed.) [1]

• Solves any 2 of the 3 equations mentioned in step 2 to get x = −3 and y = 4. Concludes that the centre of the circle is O(−3, 4). [1.5]

• ii) Substitutes the value of x and y in any one of the equations in step 1 to find the radius of the circle as: OP = √(9 − 6 + 16 − 40 + 26) = √5 units [1]

Q27 3 marks Written p. 60

Answer the questions based on the given information.Nidhi and Shikha have planned to meet at a park. Nidhi's house is at point A, and the park is at point B, as shown in the below figure.[Figure: a line graph with A(−2, 3) labelled "Nidhi's house" and B(1, 6) labelled "Park" plotted on it.]Shikha's house is at point C, the coordinates of which are unknown. Points A, B, and C lie on a straight line. The park divides the line connecting their houses such that AB:BC = 3:2.Find the coordinates of Shikha's house.

Reveal official answer

• Uses the section formula by considering C(m, n) and dividing line AC such that AB:BC = 3:2 to write: B(1, 6) = ((3×m + 2×(−2))/(3 + 2), (3×n + 2×3)/(3 + 2)) [1]

• Simplifies the expressions obtained above to form pairs of equations as (3m − 4)/5 = 1 and (3n + 6)/5 = 6. [1] [garbled in official PDF; reconstructed from the rows above and below]

• Solves the above system of equations to obtain 3m = 9 and 3n = 24 to find m = 3 and n = 8. Hence obtains the coordinates of Shikha's house as C(3, 8). [1]

Q28 1 mark Written p. 60

Answer the questions based on the given information.Nidhi and Shikha have planned to meet at a park. Nidhi's house is at point A, and the park is at point B, as shown in the below figure.[Figure: a line graph with A(−2, 3) labelled "Nidhi's house" and B(1, 6) labelled "Park" plotted on it.]Shikha's house is at point C, the coordinates of which are unknown. Points A, B, and C lie on a straight line. The park divides the line connecting their houses such that AB:BC = 3:2.Find the distance between Nidhi's house and the park.

Reveal official answer

• Uses the distance formula to find the distance between Nidhi's house and the park as: √(1 − (−2))² + (6 − 3)² = √18 = 3√2 units [1]

Q29 1 mark Written p. 60

Answer the questions based on the given information.Nidhi and Shikha have planned to meet at a park. Nidhi's house is at point A, and the park is at point B, as shown in the below figure.[Figure: a line graph with A(−2, 3) labelled "Nidhi's house" and B(1, 6) labelled "Park" plotted on it.]Shikha's house is at point C, the coordinates of which are unknown. Points A, B, and C lie on a straight line. The park divides the line connecting their houses such that AB:BC = 3:2.Find the distance between Nidhi's house and Shikha's house.

Reveal official answer

• Writes coordinates of Nidhi's house as A(−2, 3) and Shikha's house as C(3, 8). Uses the distance formula to find the distance between their houses as √(3 − (−2))² + (8 − 3)² = √50 units = 5√2 units [1]