EduSahara™ Assignment
Name : Geometric Figure Properties and Collinearity
Chapter : Distance and Section Formulae
Grade : ICSE Grade X
License : Non Commercial Use
Question 1
1.
    • Find the area of the triangle
    • formed by the points
    • (
      (
      −
      6
      )
      ,
      8
      )
    • ,
    • (
      (
      −
      3
      )
      ,
      (
      −
      2
      )
      )
    • and
    • (
      8
      ,
      6
      )
  • (i)
    67
  • (ii)
    69
  • (iii)
    65
  • (iv)
    68
  • (v)
    66
Question 2
2.
    • Find the perimeter of the triangle formed by the points
    • (
      7
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      1
      ,
      (
      −
      1
      )
      )
    • and
    • (
      (
      −
      3
      )
      ,
      (
      −
      7
      )
      )
  • (i)
    (

    √

    37
    +
    2

    √

    10
    +
    5

    √

    5
    )
  • (ii)
    (

    √

    37
    +
    2

    √

    15
    +
    5

    √

    5
    )
  • (iii)
    (

    √

    37
    +
    2

    √

    13
    +
    25
    )
  • (iv)
    (

    √

    37
    +
    2

    √

    13
    +
    5
    4
    √

    5
    )
  • (v)
    (

    √

    37
    +
    2

    √

    13
    +
    5

    √

    5
    )
Question 3
3.
    • Find the lengths of the sides of the triangle formed by the points
    • (
      1
      ,
      (
      −
      8
      )
      )
    • ,
    • (
      (
      −
      2
      )
      ,
      7
      )
    • and
    • (
      2
      ,
      (
      −
      8
      )
      )
  • (i)
      • 3

        √

        26
      • ,
      		•

        √

        241
      • ,
      		•
        4
  • (ii)
      • 3

        √

        26
      • ,
      		•
        241
      • ,
      		•
        1
  • (iii)
      • 3

        √

        26
      • ,
      		•

        √

        241
      • ,
      		•
        1
  • (iv)
      • 3
        4
        √

        26
      • ,
      		•

        √

        241
      • ,
      		•
        1
Question 4
4.
    • The points
    • (
      (
      −
      4
      )
      ,
      7
      )
    • ,
    • (
      1
      ,
      (
      −
      2
      )
      )
    • and
    • (
      (
      −
      3
      2
      +
      9
      2

      √

      3
      )
      ,
      (
      5
      2
      +
      5
      2

      √

      3
      )
      )
    • represent
  • (i)
    isosceles triangle
  • (ii)
    scalene triangle
  • (iii)
    right angled triangle
  • (iv)
    equilateral triangle
Question 5
5.
    • The points
    • (
      0
      ,
      8
      )
    • ,
    • (
      7
      ,
      6
      )
    • and
    • (
      75
      14
      ,
      27
      2
      )
    •  
    • represent
  • (i)
    collinear points
  • (ii)
    scalene triangle
  • (iii)
    right angle triangle
  • (iv)
    equilateral triangle
  • (v)
    isosceles triangle
Question 6
6.
    • The points
    • (
      (
      −
      2
      )
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      1
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      4
      ,
      (
      −
      5
      )
      )
    • and
    • (
      1
      ,
      (
      −
      8
      )
      )
    •  
    • represent
  • (i)
    square
  • (ii)
    parallelogram
  • (iii)
    rectangle
  • (iv)
    trapezium
  • (v)
    rhombus
Question 7
7.
    • The points
    • (
      (
      −
      6
      )
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      1
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      3
      ,
      1
      )
    • and
    • (
      (
      −
      4
      )
      ,
      1
      )
    • represent
  • (i)
    rhombus
  • (ii)
    parallelogram
  • (iii)
    rectangle
  • (iv)
    trapezium
  • (v)
    square
Question 8
8.
    • The points
    • (
      (
      −
      4
      )
      ,
      (
      −
      6
      )
      )
    • ,
    • (
      3
      ,
      (
      −
      6
      )
      )
    • ,
    • (
      3
      ,
      (
      −
      3
      )
      )
    • and
    • (
      (
      −
      4
      )
      ,
      (
      −
      3
      )
      )
    • represents
  • (i)
    trapezium
  • (ii)
    rectangle
  • (iii)
    rhombus
  • (iv)
    parallelogram
  • (v)
    square
Question 9
9.
    • The points
    • (
      7
      ,
      1
      )
    • ,
    • (
      6
      ,
      (
      −
      3
      )
      )
    • and
    • (
      (
      −
      1
      )
      ,
      (
      −
      6
      )
      )
    • represent
  • (i)
    equilateral triangle
  • (ii)
    scalene triangle
  • (iii)
    right angle triangle
  • (iv)
    isosceles triangle
  • (v)
    collinear points
Question 10
10.
    • The points
    • (
      (
      −
      4
      )
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      0
      ,
      2
      )
    • and
    • (
      3
      ,
      (
      −
      2
      )
      )
    • represent
  • (i)
    equilateral triangle
  • (ii)
    isosceles right angled triangle
  • (iii)
    scalene triangle
  • (iv)
    right angle triangle
  • (v)
    collinear points
Question 11
11.
    • The points
    • (
      (
      −
      4
      )
      ,
      (
      −
      4
      )
      )
    • ,
    • (
      2
      ,
      (
      −
      4
      )
      )
    • and
    • (
      2
      ,
      (
      −
      1
      )
      )
    • represent
  • (i)
    equilateral triangle
  • (ii)
    right angle triangle
  • (iii)
    scalene triangle
  • (iv)
    collinear points
  • (v)
    isosceles right angled triangle
Question 12
12.
    • The points
    • (
      (
      −
      5
      )
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      0
      ,
      (
      −
      7
      )
      )
    • ,
    • (
      5
      ,
      (
      −
      2
      )
      )
    • and
    • (
      0
      ,
      3
      )
    •  
    • represent
  • (i)
    rhombus
  • (ii)
    trapezium
  • (iii)
    parallelogram
  • (iv)
    rectangle
  • (v)
    square
Question 13
13.
    • The points
    • (
      7
      ,
      0
      )
    • ,
    • (
      (
      −
      7
      )
      ,
      0
      )
    • and
    • (
      0
      ,
      0
      )
    • represents
  • (i)
    collinear points
  • (ii)
    equilateral triangle
  • (iii)
    right angle triangle
  • (iv)
    isoceles triangle
Question 14
14.
    • Find the value of k such that the points
    • (
      (
      −
      7
      )
      ,
      (
      −
      8
      )
      )
    • ,
    • (
      (
      −
      3
      2
      )
      ,
      1
      2
      )
    • and
    • (k,9)
    • are collinear
  • (i)
    1
  • (ii)
    7
  • (iii)
    4
  • (iv)
    5
  • (v)
    3
Question 15
15.
    • Find the value of k such that the points
    • (
      (
      −
      8
      )
      ,
      (
      −
      4
      )
      )
    • ,
    • (
      (
      −
      29
      4
      )
      ,
      (
      −
      1
      )
      )
    • and
    • (-6,k)
    • are collinear
  • (i)
    3
  • (ii)
    4
  • (iii)
    5
  • (iv)
    7
  • (v)
    2
Question 16
16.
    • Two vertices of a triangle are
    • (
      0
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      2
      ,
      (
      −
      5
      )
      )
    • and its centriod is
    • (
      7
      3
      ,
      (
      −
      3
      )
      )
    • . Find the coordinates of the third vertex of the triangle
  • (i)
    (
    5
    ,
    2
    )
  • (ii)
    (
    (
    −
    2
    )
    ,
    5
    )
  • (iii)
    (
    5
    ,
    (
    −
    2
    )
    )
  • (iv)
    (
    (
    −
    5
    )
    ,
    (
    −
    2
    )
    )
  • (v)
    (
    (
    −
    5
    )
    ,
    2
    )
Question 17
17.
    • The points B
    • (
      (
      −
      1
      )
      ,
      8
      )
    • and D
    • (
      (
      −
      8
      )
      ,
      7
      )
    • are the opposite vertices
    • of a square ABCD. Find the other two vertices
  • (i)
    • (
      (
      −
      6
      )
      ,
      2
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      11
      )
  • (ii)
    • (
      (
      −
      4
      )
      ,
      4
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      11
      )
  • (iii)
    • (
      (
      −
      3
      )
      ,
      3
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      11
      )
  • (iv)
    • (
      (
      −
      4
      )
      ,
      4
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      12
      )
  • (v)
    • (
      (
      −
      2
      )
      ,
      6
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      11
      )
Question 18
18.
    • Find the coordinates of the midpoints of the sides of the quadrilateral
    • formed by
    • (
      (
      −
      4
      )
      ,
      (
      −
      4
      )
      )
    • ,
    • (
      0
      ,
      1
      )
    • ,
    • (
      (
      −
      4
      )
      ,
      2
      )
    • and
    • (
      (
      −
      6
      )
      ,
      1
      )
  • (i)
    • (
      (
      −
      2
      )
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      (
      −
      2
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      (
      −
      1
      2
      )
      )
  • (ii)
    • (
      (
      −
      2
      )
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      (
      −
      2
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      7
      )
      ,
      (
      −
      1
      2
      )
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      (
      −
      3
      2
      )
      )
  • (iii)
    • (
      (
      −
      2
      )
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      (
      −
      2
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      4
      )
      ,
      1
      2
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      (
      −
      3
      2
      )
      )
  • (iv)
    • (
      (
      −
      2
      )
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      (
      −
      2
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      (
      −
      3
      2
      )
      )
  • (v)
    • (
      0
      ,
      1
      2
      )
    • ,
    • (
      (
      −
      2
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      (
      −
      3
      2
      )
      )
Question 19
19.
    • Find the lengths of the medians of a triangle whose vertices are
    • (
      (
      −
      6
      )
      ,
      6
      )
    • ,
    • (
      1
      ,
      3
      )
    • and
    • (
      0
      ,
      (
      −
      4
      )
      )
  • (i)
      • 5
        2

        √

        2
      • ,
      		•

        √

        34
      • ,
      		•
        1
        2

        √

        58
  • (ii)
      • 13
        2

        √

        2
      • ,
      		•
        2

        √

        5
      • ,
      		•
        1
        2

        √

        314
  • (iii)
      • 1
        2

        √

        58
      • ,
      		•
        5
        2

        √

        2
      • ,
      		•
        5
        2

        √

        2
Question 20
20.
    • Find the area of the quadrilateral formed by
    • (
      (
      −
      6
      )
      ,
      (
      −
      6
      )
      )
    • ,
    • (
      (
      −
      1
      )
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      0
      )
    • and
    • (
      (
      −
      7
      )
      ,
      (
      −
      2
      )
      )
  • (i)
    19
  • (ii)
    17
  • (iii)
    18
  • (iv)
    15
  • (v)
    20
Question 21
21.
    • Three vertices of a parallelogram are
    • (
      (
      −
      5
      )
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      1
      ,
      (
      −
      5
      )
      )
    • and
    • (
      5
      ,
      (
      −
      1
      )
      )
    • .
    • Find the fourth vertex
  • (i)
    (
    1
    ,
    1
    )
  • (ii)
    (
    (
    −
    1
    )
    ,
    (
    −
    1
    )
    )
  • (iii)
    (
    (
    −
    2
    )
    ,
    0
    )
  • (iv)
    (
    0
    ,
    (
    −
    2
    )
    )
  • (v)
    (
    (
    −
    3
    )
    ,
    (
    −
    3
    )
    )
Question 22
22.
    • Find the coordinates of the orthocentre of the triangle
    • whose vertices are
    • (
      (
      −
      8
      )
      ,
      7
      )
    • ,
    • (
      5
      ,
      (
      −
      6
      )
      )
    • and
    • (
      0
      ,
      (
      −
      5
      )
      )
  • (i)
    (
    (
    −
    11
    )
    ,
    (
    −
    16
    )
    )
  • (ii)
    (
    (
    −
    12
    )
    ,
    (
    −
    19
    )
    )
  • (iii)
    (
    (
    −
    13
    )
    ,
    (
    −
    18
    )
    )
  • (iv)
    (
    (
    −
    15
    )
    ,
    (
    −
    20
    )
    )
  • (v)
    (
    (
    −
    14
    )
    ,
    (
    −
    17
    )
    )
Question 23
23.
    • Find the coordinates of the circumcentre of the triangle
    • whose vertices are
    • (
      4
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      (
      −
      2
      )
      )
    • and
    • (
      (
      −
      3
      )
      ,
      0
      )
  • (i)
    (
    (
    −
    3
    2
    )
    ,
    (
    −
    7
    2
    )
    )
  • (ii)
    (
    (
    −
    5
    2
    )
    ,
    (
    −
    13
    2
    )
    )
  • (iii)
    (
    1
    2
    ,
    (
    −
    11
    2
    )
    )
  • (iv)
    (
    3
    2
    ,
    (
    −
    5
    2
    )
    )
  • (v)
    (
    (
    −
    1
    2
    )
    ,
    (
    −
    9
    2
    )
    )
Question 24
24.
    • Find the centre of the circle passing through
    • the points
    • (
      7
      ,
      (
      −
      6
      )
      )
    • ,
    • (
      8
      ,
      6
      )
    • and
    • (
      1
      ,
      2
      )
  • (i)
    (
    27
    4
    ,
    1
    16
    )
  • (ii)
    (
    23
    4
    ,
    17
    16
    )
  • (iii)
    (
    31
    4
    ,
    (
    −
    15
    16
    )
    )
  • (iv)
    (
    19
    4
    ,
    (
    −
    31
    16
    )
    )
  • (v)
    (
    35
    4
    ,
    33
    16
    )
Question 25
25.
    • Find the circumradius of the triangle whose vertices are
    • (
      (
      −
      3
      )
      ,
      7
      )
    • ,
    • (
      (
      −
      3
      )
      ,
      4
      )
    • and
    • (
      1
      ,
      2
      )
  • (i)
    1
    4

    √

    205
  • (ii)
    1
    4

    √

    202
  • (iii)
    1
    4
    4
    √

    205
  • (iv)
    205
    4
  • (v)
    1
    4

    √

    207
Question 26
26.
    • Find the radius of the circle passing through the points
    • (
      5
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      1
      )
    • and
    • (
      (
      −
      3
      )
      ,
      (
      −
      6
      )
      )
  • (i)
    1
    118
    4
    √

    591890
  • (ii)
    295945
    59
  • (iii)
    1
    118

    √

    591890
  • (iv)
    1
    118

    √

    591892
  • (v)
    1
    118

    √

    591887
Question 27
27.
    • Find the point which is equidistant from the points
    • (
      (
      −
      3
      )
      ,
      5
      )
    • ,
    • (
      6
      ,
      (
      −
      3
      )
      )
    • and
    • (
      (
      −
      6
      )
      ,
      (
      −
      2
      )
      )
  • (i)
    (
    (
    −
    5
    6
    )
    ,
    1
    2
    )
  • (ii)
    (
    7
    6
    ,
    (
    −
    3
    2
    )
    )
  • (iii)
    (
    (
    −
    11
    6
    )
    ,
    (
    −
    5
    2
    )
    )
  • (iv)
    (
    1
    6
    ,
    (
    −
    1
    2
    )
    )
  • (v)
    (
    13
    6
    ,
    3
    2
    )
Question 28
28.
    • Find the relation between x and y if the points
    • (
      x
      ,
      y
      )
    • ,
    • (
      7
      ,
      7
      )
    • and
    • (
      4
      ,
      3
      )
    • are collinear
  • (i)
    (
    −
    11
    x
    +
    13
    y
    −
    14
    		)
    =
    0
  • (ii)
    (
    −
    4
    x
    +
    3
    y
    +
    7
    		)
    =
    0
  • (iii)
    (
    12
    x
    −
    10
    y
    +
    32
    		)
    =
    0
  • (iv)
    (
    5
    x
    −
    20
    		)
    =
    0
  • (v)
    (
    −
    7
    x
    +
    10
    y
    −
    2
    		)
    =
    0
Question 29
29.
Which of the following sets of points are collinear?
  • (i)
    (
    0
    ,
    7
    )
    ,
    (
    7
    ,
    (
    −
    7
    )
    )
    ,
    (
    4
    ,
    (
    −
    4
    )
    )
  • (ii)
    (
    (
    −
    8
    )
    ,
    8
    )
    ,
    (
    7
    ,
    2
    )
    ,
    (
    (
    −
    203
    16
    )
    ,
    79
    8
    )
  • (iii)
    (
    (
    −
    8
    )
    ,
    (
    −
    4
    )
    )
    ,
    (
    (
    −
    4
    )
    ,
    (
    −
    3
    )
    )
    ,
    (
    (
    −
    2
    )
    ,
    (
    −
    1
    )
    )
  • (iv)
    (
    (
    −
    5
    )
    ,
    (
    −
    1
    )
    )
    ,
    (
    8
    ,
    (
    −
    2
    )
    )
    ,
    (
    3
    ,
    4
    )
  • (v)
    (
    6
    ,
    (
    −
    3
    )
    )
    ,
    (
    (
    −
    3
    )
    ,
    (
    −
    4
    )
    )
    ,
    (
    (
    −
    8
    )
    ,
    2
    )
Question 30
30.
    • Find the relation between x and y such that the point P
    • (
      x
      ,
      y
      )
    • is equidistant from points
    • (
      (
      −
      2
      )
      ,
      0
      )
    • and
    • (
      8
      ,
      4
      )
  • (i)
    (
    4
    x
    +
    4
    y
    +
    8
    		)
    =
    0
  • (ii)
    (
    4
    x
    +
    y
    +
    8
    		)
    =
    0
  • (iii)
    (
    4
    x
    +
    2
    y
    −
    19
    		)
    =
    0
  • (iv)
    (
    5
    x
    +
    2
    y
    −
    19
    		)
    =
    0
  • (v)
    (
    5
    x
    +
    5
    y
    −
    19
    		)
    =
    0
    Assignment Key
  •  1) (i)
  •  2) (v)
  •  3) (iii)
  •  4) (iv)
  •  5) (v)
  •  6) (i)
  •  7) (ii)
  •  8) (ii)
  •  9) (ii)
  •  10) (ii)
  •  11) (ii)
  •  12) (i)
  •  13) (i)
  •  14) (iii)
  •  15) (ii)
  •  16) (iii)
  •  17) (ii)
  •  18) (iv)
  •  19) (ii)
  •  20) (iii)
  •  21) (ii)
  •  22) (iii)
  •  23) (v)
  •  24) (i)
  •  25) (i)
  •  26) (iii)
  •  27) (iv)
  •  28) (ii)
  •  29) (ii)
  •  30) (iv)