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Mathematics · Textbook solutions

Coordinate Geometry

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 34 questions

7.2 Distance Formula

19 q

Solved Examples

Worked · 5
  1. 7.1 Eg.1
    Do the points (3,2)(3, 2), (−2,−3)(-2, -3) and (2,3)(2, 3) form a triangle? If so, name the type of triangle formed.
  2. 7.1 Eg.2
    Show that the points (1,7)(1, 7), (4,2)(4, 2), (−1,−1)(-1, -1) and (−4,4)(-4, 4) are the vertices of a square.
  3. 7.1 Eg.3
    Fig. 7.6 shows the arrangement of desks in a classroom. Ashima, Bharti and Camella are seated at A(3,1)(3, 1), B(6,4)(6, 4) and C(8,6)(8, 6) respectively. Do you think they are seated in a line? Give reasons for your answer.
  4. 7.1 Eg.4
    Find a relation between xx and yy such that the point (x,y)(x, y) is equidistant from the points (7,1)(7, 1) and (3,5)(3, 5).
  5. 7.1 Eg.5
    Find a point on the yy-axis which is equidistant from the points A(6,5)(6, 5) and B(−4,3)(-4, 3).

Exercise 7.1

Practice · 14
  1. Find the distance between the following pairs of points :
    Ex 7.1 Q1 (i)
    (2,3)(2, 3), (4,1)(4, 1)
  2. Ex 7.1 Q1 (ii)
    (−5,7)(-5, 7), (−1,3)(-1, 3)
  3. Ex 7.1 Q1 (iii)
    (a,b)(a, b), (−a,−b)(-a, -b)
  4. Ex 7.1 Q2
    Find the distance between the points (0,0)(0, 0) and (36,15)(36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.
  5. Ex 7.1 Q3
    Determine if the points (1,5)(1, 5), (2,3)(2, 3) and (−2,−11)(-2, -11) are collinear.
  6. Ex 7.1 Q4
    Check whether (5,−2)(5, -2), (6,4)(6, 4) and (7,−2)(7, -2) are the vertices of an isosceles triangle.
  7. Ex 7.1 Q5
    In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct.
  8. Ex 7.1 Q7
    Find the point on the xx-axis which is equidistant from (2,−5)(2, -5) and (−2,9)(-2, 9).
  9. Ex 7.1 Q8
    Find the values of yy for which the distance between the points P(2,−3)(2, -3) and Q(10,y)(10, y) is 10 units.
  10. Ex 7.1 Q9
    If Q(0,1)(0, 1) is equidistant from P(5,−3)(5, -3) and R(x,6)(x, 6), find the values of xx. Also find the distances QR and PR.
  11. Ex 7.1 Q10
    Find a relation between xx and yy such that the point (x,y)(x, y) is equidistant from the point (3,6)(3, 6) and (−3,4)(-3, 4).
  12. Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
    Ex 7.1 Q6 (i)
    (−1,−2)(-1, -2), (1,0)(1, 0), (−1,2)(-1, 2), (−3,0)(-3, 0)
  13. Ex 7.1 Q6 (ii)
    (−3,5)(-3, 5), (3,1)(3, 1), (0,3)(0, 3), (−1,−4)(-1, -4)
  14. Ex 7.1 Q6 (iii)
    (4,5)(4, 5), (7,6)(7, 6), (4,3)(4, 3), (1,2)(1, 2)

7.3 Section Formula

15 q

Solved Examples

Worked · 5
  1. 7.2 Eg.6
    Find the coordinates of the point which divides the line segment joining the points (4,−3)(4, -3) and (8,5)(8, 5) in the ratio 3:13 : 1 internally.
  2. 7.2 Eg.7
    In what ratio does the point (−4,6)(-4, 6) divide the line segment joining the points A(−6,10)(-6, 10) and B(3,−8)(3, -8)?
  3. 7.2 Eg.8
    Find the coordinates of the points of trisection (i.e., points dividing in three equal parts) of the line segment joining the points A(2,−2)(2, -2) and B(−7,4)(-7, 4).
  4. 7.2 Eg.9
    Find the ratio in which the yy-axis divides the line segment joining the points (5,−6)(5, -6) and (−1,−4)(-1, -4). Also find the point of intersection.
  5. 7.2 Eg.10
    If the points A(6,1)(6, 1), B(8,2)(8, 2), C(9,4)(9, 4) and D(p,3)(p, 3) are the vertices of a parallelogram, taken in order, find the value of pp.

Exercise 7.2

Practice · 10
  1. Ex 7.2 Q1
    Find the coordinates of the point which divides the join of (−1,7)(-1, 7) and (4,−3)(4, -3) in the ratio 2:32 : 3.
  2. Ex 7.2 Q2
    Find the coordinates of the points of trisection of the line segment joining (4,−1)(4, -1) and (−2,−3)(-2, -3).
  3. Ex 7.2 Q3
    To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in Fig. 7.12. Niharika runs 14\dfrac{1}{4}th the distance AD on the 2nd line and posts a green flag. Preet runs 15\dfrac{1}{5}th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?
  4. Ex 7.2 Q4
    Find the ratio in which the line segment joining the points (−3,10)(-3, 10) and (6,−8)(6, -8) is divided by (−1,6)(-1, 6).
  5. Ex 7.2 Q5
    Find the ratio in which the line segment joining A(1,−5)(1, -5) and B(−4,5)(-4, 5) is divided by the xx-axis. Also find the coordinates of the point of division.
  6. Ex 7.2 Q6
    If (1,2)(1, 2), (4,y)(4, y), (x,6)(x, 6) and (3,5)(3, 5) are the vertices of a parallelogram taken in order, find xx and yy.
  7. Ex 7.2 Q7
    Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2,−3)(2, -3) and B is (1,4)(1, 4).
  8. Ex 7.2 Q8
    If A and B are (−2,−2)(-2, -2) and (2,−4)(2, -4), respectively, find the coordinates of P such that AP=37AB\text{AP} = \dfrac{3}{7}\text{AB} and P lies on the line segment AB.
  9. Ex 7.2 Q9
    Find the coordinates of the points which divide the line segment joining A(−2,2)(-2, 2) and B(2,8)(2, 8) into four equal parts.
  10. Ex 7.2 Q10
    Find the area of a rhombus if its vertices are (3,0)(3, 0), (4,5)(4, 5), (−1,4)(-1, 4) and (−2,−1)(-2, -1) taken in order. [Hint : Area of a rhombus =12= \dfrac{1}{2} (product of its diagonals)]