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Class 10 Maths Chapter 7: Coordinate Geometry worksheet with answer key

A 20-mark worksheet on Coordinate Geometry for about 40 minutes of class or homework: 5 multiple-choice, 3 very short answer and 3 short answer questions, written from the NCERT chapter and checked by a second reviewer. Print the PDF for the class; the answer key carries the marking scheme. Teachers may photocopy it freely.

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Question 1
While finding the distance between two points, Aarav gets PQ² = 49. The distance PQ is
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Answer: (c) 7 units

A distance is never negative, so only the positive square root is taken: PQ = 7 units.

Question 2
The distance of the point P(−6, 4) from the y-axis is
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Answer: (a) 6 units

The distance from the y-axis is the size of the abscissa: |−6| = 6 units. (4 is the distance from the x-axis; a distance is never −6; 2√13 is OP.)

Question 3
Which of the following points lies on the x-axis?
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Answer: (b) (−3, 0)

A point on the x-axis has ordinate 0, so it has the form (x, 0): (−3, 0). (0, −3) lies on the y-axis.

Question 4
The distance of the point (−9, 12) from the origin is
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Answer: (c) 15 units

OP = √(x² + y²) = √(81 + 144) = √225 = 15. (21 and 3 add and subtract 9 and 12; 3√7 = √63 subtracts the squares.)

Question 5
A kite K is flying at the point (5, −8) on a grid whose x-axis runs along a straight canal bank. The distance of K from the x-axis is
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Answer: (b) 8 units

The distance from the x-axis is the size of the ordinate: |−8| = 8 units. (5 is the distance from the y-axis; √89 is the distance from the origin.)

Question 6 · 2 marks
P and Q are the points of trisection of AB, with P nearer to A. If P is (1, −2) and Q is (4, 2), find A and B.
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P is the mid-point of AQ, so A = (2 × 1 − 4, 2 × (−2) − 2) = (−2, −6). Q is the mid-point of PB, so B = (2 × 4 − 1, 2 × 2 − (−2)) = (7, 6).

  • A = (−2, −6), using P as the mid-point of AQ (1)
  • B = (7, 6), using Q as the mid-point of PB (1)
Question 7 · 2 marks
A point P lies in the second quadrant at a distance of 3 units from the x-axis and 4 units from the y-axis. Write the coordinates of P and find its distance from the origin.
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In the second quadrant the abscissa is negative and the ordinate positive. Distance from the y-axis is 4, so x = −4; from the x-axis 3, so y = 3. P = (−4, 3) and OP = √(16 + 9) = 5.

  • P = (−4, 3) (1)
  • OP = 5 units (1)
Question 8 · 2 marks
Find the values of y for which the distance between A(−4, y) and B(2, 1) is 10 units.
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36 + (y − 1)² = 100, so (y − 1)² = 64 and y − 1 = ±8. Hence y = 9 or y = −7; both are admissible.

  • forms 36 + (y − 1)² = 100 (1)
  • y = 9 or y = −7 (1)
Question 9 · 3 marks
Find the points on the x-axis that are at a distance of 5 units from A(4, 3). Show that one of them is the origin.
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A point on the x-axis is (x, 0). (x − 4)² + 9 = 25, so (x − 4)² = 16 and x − 4 = ±4, giving x = 8 or x = 0. The points are (8, 0) and (0, 0), and (0, 0) is the origin. Check: OA = √(16 + 9) = 5.

  • takes the point as (x, 0) and forms (x − 4)² + 9 = 25 (1)
  • x − 4 = ±4 (1)
  • (8, 0) and (0, 0); the second is the origin (1)
Question 10 · 3 marks
Find the points of trisection of the segment joining A(−5, 7) and B(4, −2). Verify, using the mid-point formula, that the point nearer to B is the mid-point of the segment joining the other point of trisection and B.
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P divides AB in 1 : 2: ((4 + 2 × (−5))/3, (−2 + 2 × 7)/3) = (−2, 4). Q divides AB in 2 : 1: ((8 − 5)/3, (−4 + 7)/3) = (1, 1). Mid-point of PB = ((−2 + 4)/2, (4 − 2)/2) = (1, 1) = Q.

  • P = (−2, 4) (1)
  • Q = (1, 1) (1)
  • mid-point of PB is (1, 1) = Q (1)
Question 11 · 3 marks
A cable-car line runs straight from station A(12, 5) to station B(4, −3). A stop P on the line is at the point (x, 2). Find the ratio in which P divides AB and the value of x.
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Take the ratio k : 1. Using the ordinate, (−3k + 5)/(k + 1) = 2, so −3k + 5 = 2k + 2 and k = 3/5: the ratio is 3 : 5. Then x = (3 × 4 + 5 × 12)/8 = 72/8 = 9. P = (9, 2).

  • sets (−3k + 5)/(k + 1) = 2 (1)
  • k = 3/5, ratio 3 : 5 (1)
  • x = 9 (1)
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