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
    • (
      8
      ,
      (
      −
      6
      )
      )
    • ,
    • (
      5
      ,
      (
      −
      6
      )
      )
    • and
    • (
      (
      −
      3
      )
      ,
      (
      −
      4
      )
      )
  • (i)
    3
  • (ii)
    1
  • (iii)
    4
  • (iv)
    2
  • (v)
    6
Question 2
2.
    • Find the perimeter of the triangle formed by the points
    • (
      6
      ,
      (
      −
      3
      )
      )
    • ,
    • (
      8
      ,
      0
      )
    • and
    • (
      (
      −
      2
      )
      ,
      6
      )
  • (i)
    (

    √

    13
    +
    2

    √

    34
    +
    145
    )
  • (ii)
    (

    √

    13
    +
    2

    √

    36
    +

    √

    145
    )
  • (iii)
    (

    √

    13
    +
    2

    √

    34
    +
    4
    √

    145
    )
  • (iv)
    (

    √

    13
    +
    2

    √

    34
    +

    √

    145
    )
  • (v)
    (

    √

    13
    +
    2

    √

    32
    +

    √

    145
    )
Question 3
3.
    • Find the lengths of the sides of the triangle formed by the points
    • (
      (
      −
      2
      )
      ,
      2
      )
    • ,
    • (
      1
      ,
      1
      )
    • and
    • (
      (
      −
      4
      )
      ,
      1
      )
  • (i)

      • √

        10
      • ,
      		•
        4
      • ,
      		•

        √

        5
  • (ii)

      • √

        10
      • ,
      		•
        5
      • ,
      		•

        √

        8
  • (iii)
      • 4
        √

        10
      • ,
      		•
        5
      • ,
      		•

        √

        5
  • (iv)

      • √

        10
      • ,
      		•
        5
      • ,
      		•

        √

        5
Question 4
4.
    • The points
    • (
      5
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      (
      −
      1
      )
      )
    • and
    • (
      (
      −
      1
      2
      )
      ,
      (
      −
      11
      2

      √

      3
      −
      1
      )
      )
    • represent
  • (i)
    isosceles triangle
  • (ii)
    scalene triangle
  • (iii)
    equilateral triangle
  • (iv)
    right angled triangle
Question 5
5.
    • The points
    • (
      8
      ,
      (
      −
      8
      )
      )
    • ,
    • (
      (
      −
      8
      )
      ,
      4
      )
    • and
    • (
      (
      −
      9
      )
      ,
      (
      −
      14
      )
      )
    •  
    • represent
  • (i)
    equilateral triangle
  • (ii)
    collinear points
  • (iii)
    isosceles triangle
  • (iv)
    right angle triangle
  • (v)
    scalene triangle
Question 6
6.
    • The points
    • (
      (
      −
      4
      )
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      0
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      4
      ,
      (
      −
      5
      )
      )
    • and
    • (
      0
      ,
      (
      −
      9
      )
      )
    •  
    • represent
  • (i)
    rectangle
  • (ii)
    parallelogram
  • (iii)
    trapezium
  • (iv)
    square
  • (v)
    rhombus
Question 7
7.
    • The points
    • (
      (
      −
      1
      )
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      5
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      7
      ,
      0
      )
    • and
    • (
      1
      ,
      0
      )
    • represent
  • (i)
    square
  • (ii)
    rectangle
  • (iii)
    rhombus
  • (iv)
    parallelogram
  • (v)
    trapezium
Question 8
8.
    • The points
    • (
      (
      −
      5
      )
      ,
      (
      −
      3
      )
      )
    • ,
    • (
      1
      ,
      (
      −
      3
      )
      )
    • ,
    • (
      1
      ,
      2
      )
    • and
    • (
      (
      −
      5
      )
      ,
      2
      )
    • represents
  • (i)
    rhombus
  • (ii)
    rectangle
  • (iii)
    trapezium
  • (iv)
    parallelogram
  • (v)
    square
Question 9
9.
    • The points
    • (
      (
      −
      2
      )
      ,
      4
      )
    • ,
    • (
      3
      ,
      8
      )
    • and
    • (
      4
      ,
      6
      )
    • represent
  • (i)
    collinear points
  • (ii)
    isosceles triangle
  • (iii)
    equilateral triangle
  • (iv)
    right angle triangle
  • (v)
    scalene triangle
Question 10
10.
    • The points
    • (
      (
      −
      1
      )
      ,
      (
      −
      6
      )
      )
    • ,
    • (
      3
      ,
      (
      −
      3
      )
      )
    • and
    • (
      6
      ,
      (
      −
      7
      )
      )
    • represent
  • (i)
    isosceles right angled triangle
  • (ii)
    collinear points
  • (iii)
    scalene triangle
  • (iv)
    equilateral triangle
  • (v)
    right angle triangle
Question 11
11.
    • The points
    • (
      (
      −
      5
      )
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      1
      ,
      (
      −
      5
      )
      )
    • and
    • (
      1
      ,
      0
      )
    • represent
  • (i)
    equilateral triangle
  • (ii)
    collinear points
  • (iii)
    scalene triangle
  • (iv)
    isosceles right angled triangle
  • (v)
    right angle triangle
Question 12
12.
    • The points
    • (
      (
      −
      3
      )
      ,
      (
      −
      6
      )
      )
    • ,
    • (
      2
      ,
      (
      −
      9
      )
      )
    • ,
    • (
      7
      ,
      (
      −
      6
      )
      )
    • and
    • (
      2
      ,
      (
      −
      3
      )
      )
    •  
    • represent
  • (i)
    parallelogram
  • (ii)
    square
  • (iii)
    rectangle
  • (iv)
    rhombus
  • (v)
    trapezium
Question 13
13.
    • The points
    • (
      6
      ,
      4
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      (
      −
      4
      )
      )
    • and
    • (
      6
      7
      ,
      4
      7
      )
    • represents
  • (i)
    collinear points
  • (ii)
    right angle triangle
  • (iii)
    equilateral triangle
  • (iv)
    isoceles triangle
Question 14
14.
    • Find the value of k such that the points
    • (
      (
      −
      8
      )
      ,
      (
      −
      3
      )
      )
    • ,
    • (
      (
      −
      36
      7
      )
      ,
      9
      7
      )
    • and
    • (k,9)
    • are collinear
  • (i)
    -1
  • (ii)
    -2
  • (iii)
    0
  • (iv)
    1
  • (v)
    2
Question 15
15.
    • Find the value of k such that the points
    • (
      (
      −
      7
      )
      ,
      (
      −
      8
      )
      )
    • ,
    • (
      (
      −
      61
      19
      )
      ,
      (
      −
      16
      19
      )
      )
    • and
    • (2,k)
    • are collinear
  • (i)
    12
  • (ii)
    8
  • (iii)
    9
  • (iv)
    6
  • (v)
    10
Question 16
16.
    • Two vertices of a triangle are
    • (
      (
      −
      7
      )
      ,
      (
      −
      8
      )
      )
    • ,
    • (
      3
      ,
      2
      )
    • and its centriod is
    • (
      (
      −
      5
      3
      )
      ,
      (
      −
      1
      3
      )
      )
    • . Find the coordinates of the third vertex of the triangle
  • (i)
    (
    5
    ,
    (
    −
    1
    )
    )
  • (ii)
    (
    (
    −
    1
    )
    ,
    (
    −
    5
    )
    )
  • (iii)
    (
    1
    ,
    5
    )
  • (iv)
    (
    1
    ,
    (
    −
    5
    )
    )
  • (v)
    (
    (
    −
    1
    )
    ,
    5
    )
Question 17
17.
    • The points B
    • (
      3
      ,
      (
      −
      1
      )
      )
    • and D
    • (
      (
      −
      6
      )
      ,
      0
      )
    • are the opposite vertices
    • of a square ABCD. Find the other two vertices
  • (i)
    • (
      (
      −
      4
      )
      ,
      (
      −
      7
      )
      )
    • ,
    • (
      (
      −
      1
      )
      ,
      4
      )
  • (ii)
    • (
      0
      ,
      (
      −
      3
      )
      )
    • ,
    • (
      (
      −
      1
      )
      ,
      4
      )
  • (iii)
    • (
      (
      −
      1
      )
      ,
      (
      −
      6
      )
      )
    • ,
    • (
      (
      −
      1
      )
      ,
      4
      )
  • (iv)
    • (
      (
      −
      2
      )
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      (
      −
      2
      )
      ,
      5
      )
  • (v)
    • (
      (
      −
      2
      )
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      (
      −
      1
      )
      ,
      4
      )
Question 18
18.
    • Find the coordinates of the midpoints of the sides of the quadrilateral
    • formed by
    • (
      (
      −
      1
      )
      ,
      (
      −
      3
      )
      )
    • ,
    • (
      3
      ,
      0
      )
    • ,
    • (
      (
      −
      1
      )
      ,
      2
      )
    • and
    • (
      (
      −
      2
      )
      ,
      0
      )
  • (i)
    • (
      1
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      1
      ,
      1
      )
    • ,
    • (
      (
      −
      3
      2
      )
      ,
      1
      )
    • ,
    • (
      (
      −
      5
      2
      )
      ,
      (
      −
      1
      2
      )
      )
  • (ii)
    • (
      1
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      1
      ,
      1
      )
    • ,
    • (
      (
      −
      3
      2
      )
      ,
      1
      )
    • ,
    • (
      (
      −
      3
      2
      )
      ,
      (
      −
      3
      2
      )
      )
  • (iii)
    • (
      1
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      1
      ,
      1
      )
    • ,
    • (
      (
      −
      1
      2
      )
      ,
      0
      )
    • ,
    • (
      (
      −
      3
      2
      )
      ,
      (
      −
      3
      2
      )
      )
  • (iv)
    • (
      1
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      1
      ,
      1
      )
    • ,
    • (
      (
      −
      7
      2
      )
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      (
      −
      3
      2
      )
      ,
      (
      −
      3
      2
      )
      )
  • (v)
    • (
      3
      ,
      1
      2
      )
    • ,
    • (
      1
      ,
      1
      )
    • ,
    • (
      (
      −
      3
      2
      )
      ,
      1
      )
    • ,
    • (
      (
      −
      3
      2
      )
      ,
      (
      −
      3
      2
      )
      )
Question 19
19.
    • Find the lengths of the medians of a triangle whose vertices are
    • (
      1
      ,
      (
      −
      6
      )
      )
    • ,
    • (
      1
      ,
      0
      )
    • and
    • (
      5
      ,
      3
      )
  • (i)
      • 1
        2

        √

        241
      • ,
      		•
        5
        2
      • ,
      		•
        2

        √

        13
  • (ii)
      • 3
      • ,
      		•
        5
        2
      • ,
      		•
        5
        2
  • (iii)
      • 5
        2
      • ,
      		•
        1
        2

        √

        97
      • ,
      		•
        3
Question 20
20.
    • Find the area of the quadrilateral formed by
    • (
      (
      −
      5
      )
      ,
      (
      −
      8
      )
      )
    • ,
    • (
      1
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      (
      −
      1
      )
      ,
      (
      −
      2
      )
      )
    • and
    • (
      (
      −
      5
      )
      ,
      (
      −
      2
      )
      )
  • (i)
    27
  • (ii)
    25
  • (iii)
    23
  • (iv)
    22
  • (v)
    24
Question 21
21.
    • Three vertices of a parallelogram are
    • (
      (
      −
      6
      )
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      1
      ,
      (
      −
      2
      )
      )
    • and
    • (
      3
      ,
      3
      )
    • .
    • Find the fourth vertex
  • (i)
    (
    (
    −
    5
    )
    ,
    4
    )
  • (ii)
    (
    (
    −
    3
    )
    ,
    2
    )
  • (iii)
    (
    (
    −
    2
    )
    ,
    5
    )
  • (iv)
    (
    (
    −
    4
    )
    ,
    3
    )
  • (v)
    (
    (
    −
    6
    )
    ,
    1
    )
Question 22
22.
    • Find the coordinates of the orthocentre of the triangle
    • whose vertices are
    • (
      0
      ,
      5
      )
    • ,
    • (
      3
      ,
      (
      −
      8
      )
      )
    • and
    • (
      (
      −
      1
      )
      ,
      2
      )
  • (i)
    (
    (
    −
    202
    11
    )
    ,
    (
    −
    39
    11
    )
    )
  • (ii)
    (
    (
    −
    180
    11
    )
    ,
    (
    −
    17
    11
    )
    )
  • (iii)
    (
    (
    −
    169
    11
    )
    ,
    (
    −
    28
    11
    )
    )
  • (iv)
    (
    (
    −
    158
    11
    )
    ,
    5
    11
    )
  • (v)
    (
    (
    −
    191
    11
    )
    ,
    (
    −
    6
    11
    )
    )
Question 23
23.
    • Find the coordinates of the circumcentre of the triangle
    • whose vertices are
    • (
      (
      −
      7
      )
      ,
      2
      )
    • ,
    • (
      2
      ,
      1
      )
    • and
    • (
      6
      ,
      (
      −
      4
      )
      )
  • (i)
    (
    (
    −
    5
    2
    )
    ,
    (
    −
    17
    2
    )
    )
  • (ii)
    (
    (
    −
    11
    2
    )
    ,
    (
    −
    19
    2
    )
    )
  • (iii)
    (
    (
    −
    7
    2
    )
    ,
    (
    −
    15
    2
    )
    )
  • (iv)
    (
    (
    −
    3
    2
    )
    ,
    (
    −
    11
    2
    )
    )
  • (v)
    (
    (
    −
    9
    2
    )
    ,
    (
    −
    13
    2
    )
    )
Question 24
24.
    • Find the centre of the circle passing through
    • the points
    • (
      (
      −
      8
      )
      ,
      7
      )
    • ,
    • (
      0
      ,
      (
      −
      6
      )
      )
    • and
    • (
      (
      −
      2
      )
      ,
      7
      )
  • (i)
    (
    (
    −
    5
    )
    ,
    (
    −
    3
    26
    )
    )
  • (ii)
    (
    (
    −
    4
    )
    ,
    (
    −
    29
    26
    )
    )
  • (iii)
    (
    (
    −
    3
    )
    ,
    49
    26
    )
  • (iv)
    (
    (
    −
    6
    )
    ,
    23
    26
    )
  • (v)
    (
    (
    −
    7
    )
    ,
    (
    −
    55
    26
    )
    )
Question 25
25.
    • Find the circumradius of the triangle whose vertices are
    • (
      (
      −
      1
      )
      ,
      4
      )
    • ,
    • (
      3
      ,
      6
      )
    • and
    • (
      (
      −
      8
      )
      ,
      (
      −
      6
      )
      )
  • (i)
    39485
    26
  • (ii)
    5
    26

    √

    7895
  • (iii)
    5
    26
    4
    √

    7897
  • (iv)
    5
    26

    √

    7897
  • (v)
    5
    26

    √

    7900
Question 26
26.
    • Find the radius of the circle passing through the points
    • (
      8
      ,
      6
      )
    • ,
    • (
      7
      ,
      (
      −
      7
      )
      )
    • and
    • (
      (
      −
      3
      )
      ,
      5
      )
  • (i)
    61
    71

    √

    82
  • (ii)
    61
    71

    √

    85
  • (iii)
    61
    71

    √

    87
  • (iv)
    61
    71
    4
    √

    85
  • (v)
    5185
    71
Question 27
27.
    • Find the point which is equidistant from the points
    • (
      (
      −
      1
      )
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      (
      −
      7
      )
      ,
      8
      )
    • and
    • (
      (
      −
      8
      )
      ,
      (
      −
      1
      )
      )
  • (i)
    (
    (
    −
    11
    2
    )
    ,
    25
    6
    )
  • (ii)
    (
    (
    −
    7
    2
    )
    ,
    13
    6
    )
  • (iii)
    (
    (
    −
    9
    2
    )
    ,
    19
    6
    )
  • (iv)
    (
    (
    −
    5
    2
    )
    ,
    31
    6
    )
  • (v)
    (
    (
    −
    13
    2
    )
    ,
    7
    6
    )
Question 28
28.
    • Find the relation between x and y if the points
    • (
      x
      ,
      y
      )
    • ,
    • (
      6
      ,
      0
      )
    • and
    • (
      3
      ,
      (
      −
      8
      )
      )
    • are collinear
  • (i)
    (
    9
    x
    +
    9
    y
    +
    45
    		)
    =
    0
  • (ii)
    (
    −
    2
    x
    +
    3
    y
    −
    15
    		)
    =
    0
  • (iii)
    (
    −
    8
    x
    +
    3
    y
    +
    48
    		)
    =
    0
  • (iv)
    (
    11
    x
    +
    6
    y
    +
    15
    		)
    =
    0
  • (v)
    (
    3
    x
    +
    9
    y
    −
    18
    		)
    =
    0
Question 29
29.
Which of the following sets of points are collinear?
  • (i)
    (
    8
    ,
    (
    −
    2
    )
    )
    ,
    (
    (
    −
    3
    )
    ,
    (
    −
    3
    )
    )
    ,
    (
    0
    ,
    (
    −
    3
    )
    )
  • (ii)
    (
    (
    −
    3
    )
    ,
    8
    )
    ,
    (
    (
    −
    4
    )
    ,
    4
    )
    ,
    (
    3
    ,
    (
    −
    5
    )
    )
  • (iii)
    (
    1
    ,
    (
    −
    3
    )
    )
    ,
    (
    (
    −
    1
    )
    ,
    6
    )
    ,
    (
    (
    −
    1
    3
    )
    ,
    3
    )
  • (iv)
    (
    2
    ,
    6
    )
    ,
    (
    (
    −
    2
    )
    ,
    8
    )
    ,
    (
    3
    ,
    1
    )
  • (v)
    (
    2
    ,
    4
    )
    ,
    (
    (
    −
    6
    )
    ,
    2
    )
    ,
    (
    3
    ,
    (
    −
    1
    )
    )
Question 30
30.
    • Find the relation between x and y such that the point P
    • (
      x
      ,
      y
      )
    • is equidistant from points
    • (
      7
      ,
      7
      )
    • and
    • (
      (
      −
      1
      )
      ,
      (
      −
      5
      )
      )
  • (i)
    (
    12
    y
    −
    86
    		)
    =
    0
  • (ii)
    (
    12
    y
    −
    84
    		)
    =
    0
  • (iii)
    (
    3
    x
    +
    6
    y
    −
    18
    		)
    =
    0
  • (iv)
    (
    4
    x
    +
    6
    y
    −
    18
    		)
    =
    0
  • (v)
    (
    4
    x
    +
    9
    y
    −
    18
    		)
    =
    0
    Assignment Key
  •  1) (i)
  •  2) (iv)
  •  3) (iv)
  •  4) (iii)
  •  5) (iii)
  •  6) (iv)
  •  7) (iv)
  •  8) (ii)
  •  9) (v)
  •  10) (i)
  •  11) (v)
  •  12) (iv)
  •  13) (i)
  •  14) (iii)
  •  15) (iii)
  •  16) (v)
  •  17) (v)
  •  18) (ii)
  •  19) (i)
  •  20) (v)
  •  21) (iv)
  •  22) (ii)
  •  23) (iii)
  •  24) (i)
  •  25) (iv)
  •  26) (ii)
  •  27) (iii)
  •  28) (iii)
  •  29) (iii)
  •  30) (iv)