Geometry · Textbook solutions

Co-ordinate Geometry

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 101 questions

Distance formula

8 q

Solved Examples

Worked · 8
  1. Distance formula SolvedEx.1
    Find the distance between the points P(−1-1, 1) and Q(5, −7-7).
  2. Distance formula SolvedEx.2
    Show that points A(−3-3, 2), B(1, −2-2) and C(9, −10-10) are collinear.
  3. Distance formula SolvedEx.3
    Verify, whether points P(6, −6-6), Q(3, −7-7) and R(3, 3) are collinear.
  4. Distance formula SolvedEx.4
    Show that points (1, 7), (4, 2), (−1-1, −1-1) and (−4-4, 4) are vertices of a square.
  5. Distance formula SolvedEx.5
    Find the co-ordinates of a point on Y-axis which is equidistant from M(−5-5, −2-2) and N(3, 2).
  6. Distance formula SolvedEx.6
    A(−3-3, −4-4), B(−5-5, 0), C(3, 0) are the vertices of △ABC\triangle ABC. Find the co-ordinates of the circumcentre of △ABC\triangle ABC.
  7. Distance formula SolvedEx.7
    If point (x,y)(x, y) is equidistant from points (7, 1) and (3, 5), show that y=x−2y = x - 2.
  8. Distance formula SolvedEx.8
    Find the value of yy if distance between points A(2, −2-2) and B(−1-1, yy) is 5.

Practice set 5.1

16 q
  1. Find the distance between each of the following pairs of points.
    Ex 5.1 Q.1 (1)
    A(2, 3), B(4, 1)
  2. Ex 5.1 Q.1 (2)
    P(−5-5, 7), Q(−1-1, 3)
  3. Ex 5.1 Q.1 (3)
    R(0, −3-3), S(0,52)\left(0, \dfrac{5}{2}\right)
  4. Ex 5.1 Q.1 (4)
    L(5, −8-8), M(−7-7, −3-3)
  5. Ex 5.1 Q.1 (5)
    T(−3-3, 6), R(9, −10-10)
  6. Ex 5.1 Q.1 (6)
    W(−72,4)\left(\dfrac{-7}{2}, 4\right), X(11, 4)
  7. Determine whether the points are collinear.
    Ex 5.1 Q.2 (1)
    A(1, −3-3), B(2, −5-5), C(−4-4, 7)
  8. Ex 5.1 Q.2 (2)
    L(−2-2, 3), M(1, −3-3), N(5, 4)
  9. Ex 5.1 Q.2 (3)
    R(0, 3), D(2, 1), S(3, −1-1)
  10. Ex 5.1 Q.2 (4)
    P(−2-2, 3), Q(1, 2), R(4, 1)
  11. Ex 5.1 Q.3
    Find the point on the X-axis which is equidistant from A(−3-3, 4) and B(1, −4-4).
  12. Ex 5.1 Q.4
    Verify that points P(−2-2, 2), Q(2, 2) and R(2, 7) are vertices of a right angled triangle.
  13. Ex 5.1 Q.5
    Show that points P(2, −2-2), Q(7, 3), R(11, −1-1) and S(6, −6-6) are vertices of a parallelogram.
  14. Ex 5.1 Q.6
    Show that points A(−4-4, −7-7), B(−1-1, 2), C(8, 5) and D(5, −4-4) are vertices of a rhombus ABCD.
  15. Ex 5.1 Q.7
    Find xx if distance between points L(xx, 7) and M(1, 15) is 10.
  16. Ex 5.1 Q.8
    Show that the points A(1, 2), B(1, 6), C(1+23,4)(1 + 2\sqrt{3}, 4) are vertices of an equilateral triangle.

Co-ordinates of the midpoint of a segment

2 q

Solved Examples

Worked · 2
  1. Co-ordinates of the midpoint of a segment SolvedEx.1
    If A(3, 5), B(7, 9) and point Q divides seg AB in the ratio 2 : 3 then find co-ordinates of point Q.
  2. Co-ordinates of the midpoint of a segment SolvedEx.2
    Find the co-ordinates of point P if P is the midpoint of a line segment AB with A(−4-4, 2) and B(6, 2).

Centroid formula

4 q

Solved Examples

Worked · 4
  1. Centroid formula SolvedEx.1
    If point T divides the segment AB with A(−7-7, 4) and B(−6-6, −5-5) in the ratio 7 : 2, find the co-ordinates of T.
  2. Centroid formula SolvedEx.2
    If point P(−4-4, 6) divides the line segment AB with A(−6-6, 10) and B(r, s) in the ratio 2 : 1, find the co-ordinates of B.
  3. Centroid formula SolvedEx.3
    A(15, 5), B(9, 20) and A−-P−-B. Find the ratio in which point P(11, 15) divides segment AB.
  4. Centroid formula SolvedEx.4
    Find the co-ordinates of the points of trisection of the segment joining the points A(2, −2-2) and B(−7-7, 4). (The two points that divide the line segment in three equal parts are called as points of trisection of the segment.)

Practice set 5.2

16 q
  1. Ex 5.2 Q.1
    Find the coordinates of point P if P divides the line segment joining the points A(−1-1, 7) and B(4, −3-3) in the ratio 2 : 3.
  2. In each of the following examples find the co-ordinates of point A which divides segment PQ in the ratio a:ba : b.
    Ex 5.2 Q.2 (1)
    P(−3-3, 7), Q(1, −4-4), a:b=2:1a : b = 2 : 1
  3. Ex 5.2 Q.2 (2)
    P(−2-2, −5-5), Q(4, 3), a:b=3:4a : b = 3 : 4
  4. Ex 5.2 Q.2 (3)
    P(2, 6), Q(−4-4, 1), a:b=1:2a : b = 1 : 2
  5. Ex 5.2 Q.3
    Find the ratio in which point T(−1-1, 6) divides the line segment joining the points P(−3-3, 10) and Q(6, −8-8).
  6. Ex 5.2 Q.4
    Point P is the centre of the circle and AB is a diameter. Find the coordinates of point B if coordinates of point A and P are (2, −3-3) and (−2-2, 0) respectively.
  7. Ex 5.2 Q.5
    Find the ratio in which point P(kk, 7) divides the segment joining A(8, 9) and B(1, 2). Also find kk.
  8. Ex 5.2 Q.6
    Find the coordinates of midpoint of the segment joining the points (22, 20) and (0, 16).
  9. Find the centroids of the triangles whose vertices are given below.
    Ex 5.2 Q.7 (1)
    (−7-7, 6), (2, −2-2), (8, 5)
  10. Ex 5.2 Q.7 (2)
    (3, −5-5), (4, 3), (11, −4-4)
  11. Ex 5.2 Q.7 (3)
    (4, 7), (8, 4), (7, 11)
  12. Ex 5.2 Q.8
    In △ABC\triangle ABC, G(−4-4, −7-7) is the centroid. If A(−14-14, −19-19) and B(3, 5) then find the co-ordinates of C.
  13. Ex 5.2 Q.9
    A(hh, −6-6), B(2, 3) and C(−6-6, kk) are the co-ordinates of vertices of a triangle whose centroid is G(1, 5). Find hh and kk.
  14. Ex 5.2 Q.10
    Find the co-ordinates of the points of trisection of the line segment AB with A(2, 7) and B(−4-4, −8-8).
  15. Ex 5.2 Q.11
    If A(−14-14, −10-10), B(6, −2-2) is given, find the coordinates of the points which divide segment AB into four equal parts.
  16. Ex 5.2 Q.12
    If A(20, 10), B(0, 20) are given, find the coordinates of the points which divide segment AB into five congruent parts.

Slope of a line

4 q

Solved Examples

Worked · 4
  1. Slope of a line SolvedEx.1
    Find the slope of the line passing through the points A(−3-3, 5) and B(4, −1-1).
  2. Slope of a line SolvedEx.2
    Show that points P(−2-2, 3), Q(1, 2), R(4, 1) are collinear.
  3. Slope of a line SolvedEx.3
    If slope of the line joining points P(k, 0) and Q(−3-3, −2-2) is 27\dfrac{2}{7} then find kk.
  4. Slope of a line SolvedEx.4
    If A(6, 1), B(8, 2), C(9, 4) and D(7, 3) are the vertices of □ABCD\square ABCD, show that □ABCD\square ABCD is a parallelogram.

Practice set 5.3

20 q
  1. Angles made by the line with the positive direction of X-axis are given. Find the slope of these lines.
    Ex 5.3 Q.1 (1)
    45∘45^\circ
  2. Ex 5.3 Q.1 (2)
    60∘60^\circ
  3. Ex 5.3 Q.1 (3)
    90∘90^\circ
  4. Find the slopes of the lines passing through the given points.
    Ex 5.3 Q.2 (1)
    A(2, 3), B(4, 7)
  5. Ex 5.3 Q.2 (2)
    P(−3-3, 1), Q(5, −2-2)
  6. Ex 5.3 Q.2 (3)
    C(5, −2-2), D(7, 3)
  7. Ex 5.3 Q.2 (4)
    L(−2-2, −3-3), M(−6-6, −8-8)
  8. Ex 5.3 Q.2 (5)
    E(−4-4, −2-2), F(6, 3)
  9. Ex 5.3 Q.2 (6)
    T(0, −3-3), S(0, 4)
  10. Determine whether the following points are collinear.
    Ex 5.3 Q.3 (1)
    A(−1-1, −1-1), B(0, 1), C(1, 3)
  11. Ex 5.3 Q.3 (2)
    D(−2-2, −3-3), E(1, 0), F(2, 1)
  12. Ex 5.3 Q.3 (3)
    L(2, 5), M(3, 3), N(5, 1)
  13. Ex 5.3 Q.3 (4)
    P(2, −5-5), Q(1, −3-3), R(−2-2, 3)
  14. Ex 5.3 Q.3 (5)
    R(1, −4-4), S(−2-2, 2), T(−3-3, 4)
  15. Ex 5.3 Q.3 (6)
    A(−4-4, 4), K(−2,52)\left(-2, \dfrac{5}{2}\right), N(4, −2-2)
  16. Ex 5.3 Q.4
    If A(1, −1-1), B(0, 4), C(−5-5, 3) are vertices of a triangle then find the slope of each side.
  17. Ex 5.3 Q.5
    Show that A(−4-4, −7-7), B(−1-1, 2), C(8, 5) and D(5, −4-4) are the vertices of a parallelogram.
  18. Ex 5.3 Q.6
    Find kk, if R(1, −1-1), S(−2-2, kk) and slope of line RS is −2-2.
  19. Ex 5.3 Q.7
    Find kk, if B(kk, −5-5), C(1, 2) and slope of the line is 7.
  20. Ex 5.3 Q.8
    Find kk, if PQ ∥\parallel RS and P(2, 4), Q(3, 6), R(3, 1), S(5, kk).

Problem set 5

31 q
  1. Fill in the blanks using correct alternatives.
    PS5 Q.1 (1)
    Seg AB is parallel to Y-axis and coordinates of point A are (1, 3) then co-ordinates of point B can be ........ .
    1. A.
      (3, 1)
    2. B.
      (5, 3)
    3. C.
      (3, 0)
    4. D.
      (1, −3-3)
  2. PS5 Q.1 (2)
    Out of the following, point ........ lies to the right of the origin on X-axis.
    1. A.
      (−2-2, 0)
    2. B.
      (0, 2)
    3. C.
      (2, 3)
    4. D.
      (2, 0)
  3. PS5 Q.1 (3)
    Distance of point (−3-3, 4) from the origin is ...... .
    1. A.
      7
    2. B.
      1
    3. C.
      5
    4. D.
      −5-5
  4. PS5 Q.1 (4)
    A line makes an angle of 30∘30^\circ with the positive direction of X-axis. So the slope of the line is .......... .
    1. A.
      12\dfrac{1}{2}
    2. B.
      32\dfrac{\sqrt{3}}{2}
    3. C.
      13\dfrac{1}{\sqrt{3}}
    4. D.
      3\sqrt{3}
  5. Determine whether the given points are collinear.
    PS5 Q.2 (1)
    A(0, 2), B(1, −0.5-0.5), C(2, −3-3)
  6. PS5 Q.2 (2)
    P(1, 2), Q(2,85)\left(2, \dfrac{8}{5}\right), R(3,65)\left(3, \dfrac{6}{5}\right)
  7. PS5 Q.2 (3)
    L(1, 2), M(5, 3), N(8, 6)
  8. PS5 Q.3
    Find the coordinates of the midpoint of the line segment joining P(0, 6) and Q(12, 20).
  9. PS5 Q.4
    Find the ratio in which the line segment joining the points A(3, 8) and B(−9-9, 3) is divided by the Y-axis.
  10. PS5 Q.5
    Find the point on X-axis which is equidistant from P(2, −5-5) and Q(−2-2, 9).
  11. Find the distances between the following points.
    PS5 Q.6 (i)
    A(aa, 0), B(0, aa)
  12. PS5 Q.6 (ii)
    P(−6-6, −3-3), Q(−1-1, 9)
  13. PS5 Q.6 (iii)
    R(−3a-3a, aa), S(aa, −2a-2a)
  14. PS5 Q.7
    Find the coordinates of the circumcentre of a triangle whose vertices are (−3-3, 1), (0, −2-2) and (1, 3).
  15. In the following examples, can the segment joining the given points form a triangle ? If triangle is formed, state the type of the triangle considering sides of the triangle.
    PS5 Q.8 (1)
    L(6, 4), M(−5-5, −3-3), N(−6-6, 8)
  16. PS5 Q.8 (2)
    P(−2-2, −6-6), Q(−4-4, −2-2), R(−5-5, 0)
  17. PS5 Q.8 (3)
    A(2,2)(\sqrt{2}, \sqrt{2}), B(−2,−2)(-\sqrt{2}, -\sqrt{2}), C(−6,6)(-\sqrt{6}, \sqrt{6})
  18. PS5 Q.9
    Find kk if the line passing through points P(−12-12, −3-3) and Q(4, kk) has slope 12\dfrac{1}{2}.
  19. PS5 Q.10
    Show that the line joining the points A(4, 8) and B(5, 5) is parallel to the line joining the points C(2, 4) and D(1, 7).
  20. PS5 Q.11
    Show that points P(1, −2-2), Q(5, 2), R(3, −1-1), S(−1-1, −5-5) are the vertices of a parallelogram.
  21. PS5 Q.12
    Show that the □\square PQRS formed by P(2, 1), Q(−1-1, 3), R(−5-5, −3-3) and S(−2-2, −5-5) is a rectangle.
  22. PS5 Q.13
    Find the lengths of the medians of a triangle whose vertices are A(−1-1, 1), B(5, −3-3) and C(3, 5).
  23. PS5 Q.14
    Find the coordinates of centroid of the triangles if points D(−7-7, 6), E(8, 5) and F(2, −2-2) are the mid points of the sides of that triangle. [Note: marked ⋆\star (challenging) in the textbook.]
  24. PS5 Q.15
    Show that A(4, −1-1), B(6, 0), C(7, −2-2) and D(5, −3-3) are vertices of a square.
  25. PS5 Q.16
    Find the coordinates of circumcentre and radius of circumcircle of △ABC\triangle ABC if A(7, 1), B(3, 5) and C(2, 0) are given.
  26. PS5 Q.17
    Given A(4, −3-3), B(8, 5). Find the coordinates of the point that divides segment AB in the ratio 3 : 1.
  27. PS5 Q.18
    Find the type of the quadrilateral if points A(−4-4, −2-2), B(−3-3, −7-7), C(3, −2-2) and D(2, 3) are joined serially. [Note: marked ⋆\star (challenging) in the textbook.]
  28. PS5 Q.19
    The line segment AB is divided into five congruent parts at P, Q, R and S such that A−-P−-Q−-R−-S−-B. If point Q(12, 14) and S(4, 18) are given find the coordinates of A, P, R, B. [Note: marked ⋆\star (challenging) in the textbook.]
  29. PS5 Q.20
    Find the coordinates of the centre of the circle passing through the points P(6, −6-6), Q(3, −7-7) and R(3, 3).
  30. PS5 Q.21
    Find the possible pairs of coordinates of the fourth vertex D of the parallelogram, if three of its vertices are A(5, 6), B(1, −2-2) and C(3, −2-2). [Note: marked ⋆\star (challenging) in the textbook.]
  31. PS5 Q.22
    Find the slope of the diagonals of a quadrilateral with vertices A(1, 7), B(6, 3), C(0, −3-3) and D(−3-3, 3).