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
    • (
      3
      ,
      (
      −
      3
      )
      )
    • ,
    • (
      (
      −
      3
      )
      ,
      5
      )
    • and
    • (
      8
      ,
      (
      −
      1
      )
      )
  • (i)
    28
  • (ii)
    25
  • (iii)
    27
  • (iv)
    26
  • (v)
    23
Question 2
2.
    • Find the perimeter of the triangle formed by the points
    • (
      (
      −
      6
      )
      ,
      (
      −
      8
      )
      )
    • ,
    • (
      0
      ,
      (
      −
      7
      )
      )
    • and
    • (
      2
      ,
      0
      )
  • (i)
    (

    √

    37
    +

    √

    53
    +
    16
    )
  • (ii)
    (

    √

    37
    +

    √

    53
    +
    8

    √

    2
    )
  • (iii)
    (

    √

    37
    +

    √

    50
    +
    8

    √

    2
    )
  • (iv)
    (

    √

    37
    +

    √

    53
    +
    8
    4
    √

    2
    )
  • (v)
    (

    √

    37
    +

    √

    56
    +
    8

    √

    2
    )
Question 3
3.
    • Find the lengths of the sides of the triangle formed by the points
    • (
      (
      −
      7
      )
      ,
      5
      )
    • ,
    • (
      7
      ,
      3
      )
    • and
    • (
      (
      −
      8
      )
      ,
      (
      −
      7
      )
      )
  • (i)
      • 10

        √

        2
      • ,
      		•
        5

        √

        13
      • ,
      		•

        √

        145
  • (ii)
      • 10
        4
        √

        2
      • ,
      		•
        5

        √

        13
      • ,
      		•

        √

        145
  • (iii)
      • 10

        √

        2
      • ,
      		•
        65
      • ,
      		•

        √

        145
  • (iv)
      • 10

        √

        2
      • ,
      		•
        5

        √

        13
      • ,
      		•

        √

        148
Question 4
4.
    • The points
    • (
      (
      −
      4
      )
      ,
      7
      )
    • ,
    • (
      (
      −
      8
      )
      ,
      (
      −
      8
      )
      )
    • and
    • (
      (
      −
      6
      +
      15
      2

      √

      3
      )
      ,
      (
      −
      1
      2
      −
      2

      √

      3
      )
      )
    • represent
  • (i)
    equilateral triangle
  • (ii)
    isosceles triangle
  • (iii)
    right angled triangle
  • (iv)
    scalene triangle
Question 5
5.
    • The points
    • (
      8
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      2
      ,
      (
      −
      6
      )
      )
    • and
    • (
      65
      7
      ,
      (
      −
      73
      7
      )
      )
    •  
    • represent
  • (i)
    right angle triangle
  • (ii)
    equilateral triangle
  • (iii)
    scalene triangle
  • (iv)
    isosceles triangle
  • (v)
    collinear points
Question 6
6.
    • The points
    • (
      (
      −
      5
      )
      ,
      (
      −
      6
      )
      )
    • ,
    • (
      (
      −
      2
      )
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      2
      ,
      (
      −
      5
      )
      )
    • and
    • (
      (
      −
      1
      )
      ,
      (
      −
      9
      )
      )
    •  
    • represent
  • (i)
    parallelogram
  • (ii)
    square
  • (iii)
    rhombus
  • (iv)
    trapezium
  • (v)
    rectangle
Question 7
7.
    • The points
    • (
      (
      −
      5
      )
      ,
      (
      −
      4
      )
      )
    • ,
    • (
      2
      ,
      (
      −
      4
      )
      )
    • ,
    • (
      6
      ,
      (
      −
      1
      )
      )
    • and
    • (
      (
      −
      1
      )
      ,
      (
      −
      1
      )
      )
    • represent
  • (i)
    parallelogram
  • (ii)
    rectangle
  • (iii)
    square
  • (iv)
    trapezium
  • (v)
    rhombus
Question 8
8.
    • The points
    • (
      (
      −
      5
      )
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      3
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      3
      ,
      2
      )
    • and
    • (
      (
      −
      5
      )
      ,
      2
      )
    • represents
  • (i)
    rhombus
  • (ii)
    trapezium
  • (iii)
    rectangle
  • (iv)
    square
  • (v)
    parallelogram
Question 9
9.
    • The points
    • (
      8
      ,
      8
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      4
      )
    • and
    • (
      4
      ,
      0
      )
    • represent
  • (i)
    right angle triangle
  • (ii)
    equilateral triangle
  • (iii)
    collinear points
  • (iv)
    scalene triangle
  • (v)
    isosceles triangle
Question 10
10.
    • The points
    • (
      (
      −
      4
      )
      ,
      (
      −
      4
      )
      )
    • ,
    • (
      0
      ,
      (
      −
      1
      )
      )
    • and
    • (
      3
      ,
      (
      −
      5
      )
      )
    • represent
  • (i)
    equilateral triangle
  • (ii)
    isosceles right angled triangle
  • (iii)
    scalene triangle
  • (iv)
    collinear points
  • (v)
    right angle triangle
Question 11
11.
    • The points
    • (
      (
      −
      6
      )
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      2
      ,
      (
      −
      1
      )
      )
    • and
    • (
      2
      ,
      2
      )
    • represent
  • (i)
    equilateral triangle
  • (ii)
    collinear points
  • (iii)
    isosceles right angled triangle
  • (iv)
    scalene triangle
  • (v)
    right angle triangle
Question 12
12.
    • The points
    • (
      (
      −
      4
      )
      ,
      (
      −
      4
      )
      )
    • ,
    • (
      3
      ,
      (
      −
      9
      )
      )
    • ,
    • (
      10
      ,
      (
      −
      4
      )
      )
    • and
    • (
      3
      ,
      1
      )
    •  
    • represent
  • (i)
    trapezium
  • (ii)
    rhombus
  • (iii)
    parallelogram
  • (iv)
    square
  • (v)
    rectangle
Question 13
13.
    • The points
    • (
      (
      −
      6
      )
      ,
      5
      )
    • ,
    • (
      6
      ,
      (
      −
      5
      )
      )
    • and
    • (
      3
      2
      ,
      (
      −
      5
      4
      )
      )
    • represents
  • (i)
    equilateral triangle
  • (ii)
    collinear points
  • (iii)
    right angle triangle
  • (iv)
    isoceles triangle
Question 14
14.
    • Find the value of k such that the points
    • (
      (
      −
      4
      )
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      (
      −
      17
      8
      )
      ,
      5
      4
      )
    • and
    • (k,5)
    • are collinear
  • (i)
    1
  • (ii)
    -2
  • (iii)
    4
  • (iv)
    0
  • (v)
    2
Question 15
15.
    • Find the value of k such that the points
    • (
      (
      −
      4
      )
      ,
      2
      )
    • ,
    • (
      (
      −
      1
      )
      ,
      11
      4
      )
    • and
    • (4,k)
    • are collinear
  • (i)
    4
  • (ii)
    7
  • (iii)
    3
  • (iv)
    5
  • (v)
    1
Question 16
16.
    • Two vertices of a triangle are
    • (
      (
      −
      2
      )
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      (
      −
      7
      )
      ,
      7
      )
    • and its centriod is
    • (
      (
      −
      8
      3
      )
      ,
      3
      )
    • . Find the coordinates of the third vertex of the triangle
  • (i)
    (
    3
    ,
    1
    )
  • (ii)
    (
    (
    −
    1
    )
    ,
    3
    )
  • (iii)
    (
    (
    −
    1
    )
    ,
    (
    −
    3
    )
    )
  • (iv)
    (
    1
    ,
    3
    )
  • (v)
    (
    1
    ,
    (
    −
    3
    )
    )
Question 17
17.
    • The points B
    • (
      (
      −
      1
      )
      ,
      10
      )
    • and D
    • (
      (
      −
      9
      )
      ,
      8
      )
    • are the opposite vertices
    • of a square ABCD. Find the other two vertices
  • (i)
    • (
      (
      −
      4
      )
      ,
      5
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      13
      )
  • (ii)
    • (
      (
      −
      6
      )
      ,
      3
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      13
      )
  • (iii)
    • (
      (
      −
      2
      )
      ,
      7
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      13
      )
  • (iv)
    • (
      (
      −
      4
      )
      ,
      5
      )
    • ,
    • (
      (
      −
      7
      )
      ,
      14
      )
  • (v)
    • (
      (
      −
      3
      )
      ,
      4
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      13
      )
Question 18
18.
    • Find the coordinates of the midpoints of the sides of the quadrilateral
    • formed by
    • (
      (
      −
      4
      )
      ,
      (
      −
      4
      )
      )
    • ,
    • (
      (
      −
      1
      )
      ,
      1
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      2
      )
    • and
    • (
      (
      −
      6
      )
      ,
      (
      −
      2
      )
      )
  • (i)
    • (
      (
      −
      5
      2
      )
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      (
      −
      7
      2
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      0
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      (
      −
      2
      )
      )
  • (ii)
    • (
      (
      −
      5
      2
      )
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      (
      −
      7
      2
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      8
      )
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      (
      −
      3
      )
      )
  • (iii)
    • (
      (
      −
      1
      2
      )
      ,
      1
      2
      )
    • ,
    • (
      (
      −
      7
      2
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      0
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      (
      −
      3
      )
      )
  • (iv)
    • (
      (
      −
      5
      2
      )
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      (
      −
      7
      2
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      0
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      (
      −
      3
      )
      )
  • (v)
    • (
      (
      −
      5
      2
      )
      ,
      (
      −
      3
      2
      )
      )
    • ,
    • (
      (
      −
      7
      2
      )
      ,
      3
      2
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      (
      −
      3
      )
      )
Question 19
19.
    • Find the lengths of the medians of a triangle whose vertices are
    • (
      5
      ,
      (
      −
      1
      )
      )
    • ,
    • (
      4
      ,
      (
      −
      3
      )
      )
    • and
    • (
      (
      −
      2
      )
      ,
      (
      −
      5
      )
      )
  • (i)
      • 5
      • ,
      		•
        5
        2
      • ,
      		•
        1
        2

        √

        205
  • (ii)
      • 1
        2

        √

        5
      • ,
      		•

        √

        10
      • ,
      		•

        √

        10
  • (iii)

      • √

        10
      • ,
      		•
        1
        2

        √

        65
      • ,
      		•
        1
        2

        √

        5
Question 20
20.
    • Find the area of the quadrilateral formed by
    • (
      (
      −
      5
      )
      ,
      (
      −
      2
      )
      )
    • ,
    • (
      0
      ,
      2
      )
    • ,
    • (
      (
      −
      5
      )
      ,
      4
      )
    • and
    • (
      (
      −
      6
      )
      ,
      1
      )
  • (i)
    17
  • (ii)
    19
  • (iii)
    18
  • (iv)
    21
  • (v)
    15
Question 21
21.
    • Three vertices of a parallelogram are
    • (
      (
      −
      4
      )
      ,
      (
      −
      5
      )
      )
    • ,
    • (
      3
      ,
      (
      −
      5
      )
      )
    • and
    • (
      6
      ,
      (
      −
      2
      )
      )
    • .
    • Find the fourth vertex
  • (i)
    (
    (
    −
    3
    )
    ,
    (
    −
    4
    )
    )
  • (ii)
    (
    0
    ,
    (
    −
    3
    )
    )
  • (iii)
    (
    (
    −
    1
    )
    ,
    (
    −
    2
    )
    )
  • (iv)
    (
    1
    ,
    0
    )
  • (v)
    (
    (
    −
    2
    )
    ,
    (
    −
    1
    )
    )
Question 22
22.
    • Find the coordinates of the orthocentre of the triangle
    • whose vertices are
    • (
      (
      −
      3
      )
      ,
      7
      )
    • ,
    • (
      8
      ,
      0
      )
    • and
    • (
      (
      −
      5
      )
      ,
      8
      )
  • (i)
    (
    220
    3
    ,
    401
    3
    )
  • (ii)
    (
    223
    3
    ,
    398
    3
    )
  • (iii)
    (
    217
    3
    ,
    392
    3
    )
  • (iv)
    (
    226
    3
    ,
    395
    3
    )
  • (v)
    (
    229
    3
    ,
    404
    3
    )
Question 23
23.
    • Find the coordinates of the circumcentre of the triangle
    • whose vertices are
    • (
      (
      −
      4
      )
      ,
      (
      −
      7
      )
      )
    • ,
    • (
      (
      −
      6
      )
      ,
      (
      −
      7
      )
      )
    • and
    • (
      (
      −
      1
      )
      ,
      5
      )
  • (i)
    (
    (
    −
    5
    )
    ,
    (
    −
    3
    8
    )
    )
  • (ii)
    (
    (
    −
    3
    )
    ,
    13
    8
    )
  • (iii)
    (
    (
    −
    6
    )
    ,
    5
    8
    )
  • (iv)
    (
    (
    −
    4
    )
    ,
    (
    −
    11
    8
    )
    )
  • (v)
    (
    (
    −
    7
    )
    ,
    (
    −
    19
    8
    )
    )
Question 24
24.
    • Find the centre of the circle passing through
    • the points
    • (
      2
      ,
      3
      )
    • ,
    • (
      (
      −
      8
      )
      ,
      4
      )
    • and
    • (
      (
      −
      4
      )
      ,
      (
      −
      5
      )
      )
  • (i)
    (
    (
    −
    98
    43
    )
    ,
    (
    −
    25
    86
    )
    )
  • (ii)
    (
    (
    −
    227
    43
    )
    ,
    (
    −
    111
    86
    )
    )
  • (iii)
    (
    (
    −
    141
    43
    )
    ,
    61
    86
    )
  • (iv)
    (
    (
    −
    55
    43
    )
    ,
    233
    86
    )
  • (v)
    (
    (
    −
    184
    43
    )
    ,
    147
    86
    )
Question 25
25.
    • Find the circumradius of the triangle whose vertices are
    • (
      5
      ,
      4
      )
    • ,
    • (
      (
      −
      8
      )
      ,
      (
      −
      6
      )
      )
    • and
    • (
      2
      ,
      (
      −
      2
      )
      )
  • (i)
    1
    16
    4
    √

    39005
  • (ii)
    1
    16

    √

    39007
  • (iii)
    1
    16

    √

    39002
  • (iv)
    39005
    16
  • (v)
    1
    16

    √

    39005
Question 26
26.
    • Find the radius of the circle passing through the points
    • (
      (
      −
      4
      )
      ,
      8
      )
    • ,
    • (
      1
      ,
      4
      )
    • and
    • (
      0
      ,
      (
      −
      5
      )
      )
  • (i)
    41
    98

    √

    372
  • (ii)
    41
    98
    4
    √

    370
  • (iii)
    7585
    49
  • (iv)
    41
    98

    √

    370
  • (v)
    41
    98

    √

    367
Question 27
27.
    • Find the point which is equidistant from the points
    • (
      (
      −
      8
      )
      ,
      6
      )
    • ,
    • (
      (
      −
      2
      )
      ,
      1
      )
    • and
    • (
      1
      ,
      (
      −
      7
      )
      )
  • (i)
    (
    (
    −
    1051
    66
    )
    ,
    (
    −
    163
    22
    )
    )
  • (ii)
    (
    (
    −
    853
    66
    )
    ,
    (
    −
    141
    22
    )
    )
  • (iii)
    (
    (
    −
    919
    66
    )
    ,
    (
    −
    207
    22
    )
    )
  • (iv)
    (
    (
    −
    1117
    66
    )
    ,
    (
    −
    229
    22
    )
    )
  • (v)
    (
    (
    −
    985
    66
    )
    ,
    (
    −
    185
    22
    )
    )
Question 28
28.
    • Find the relation between x and y if the points
    • (
      x
      ,
      y
      )
    • ,
    • (
      4
      ,
      5
      )
    • and
    • (
      (
      −
      4
      )
      ,
      (
      −
      5
      )
      )
    • are collinear
  • (i)
    (
    −
    10
    x
    +
    8
    y
    		)
    =
    0
  • (ii)
    (
    9
    x
    +
    16
    y
    −
    24
    		)
    =
    0
  • (iii)
    (
    11
    x
    +
    4
    y
    +
    64
    		)
    =
    0
  • (iv)
    (
    −
    8
    x
    −
    4
    y
    +
    52
    		)
    =
    0
  • (v)
    (
    2
    x
    −
    12
    y
    −
    52
    		)
    =
    0
Question 29
29.
Which of the following sets of points are collinear?
  • (i)
    (
    6
    ,
    8
    )
    ,
    (
    0
    ,
    1
    )
    ,
    (
    2
    ,
    (
    −
    5
    )
    )
  • (ii)
    (
    0
    ,
    2
    )
    ,
    (
    7
    ,
    8
    )
    ,
    (
    (
    −
    4
    )
    ,
    (
    −
    8
    )
    )
  • (iii)
    (
    3
    ,
    0
    )
    ,
    (
    4
    ,
    (
    −
    2
    )
    )
    ,
    (
    (
    −
    6
    )
    ,
    5
    )
  • (iv)
    (
    (
    −
    5
    )
    ,
    8
    )
    ,
    (
    5
    ,
    (
    −
    4
    )
    )
    ,
    (
    0
    ,
    0
    )
  • (v)
    (
    (
    −
    8
    )
    ,
    4
    )
    ,
    (
    3
    ,
    (
    −
    1
    )
    )
    ,
    (
    (
    −
    19
    12
    )
    ,
    13
    12
    )
Question 30
30.
    • Find the relation between x and y such that the point P
    • (
      x
      ,
      y
      )
    • is equidistant from points
    • (
      (
      −
      3
      )
      ,
      7
      )
    • and
    • (
      (
      −
      7
      )
      ,
      (
      −
      4
      )
      )
  • (i)
    (
    7
    x
    +
    22
    y
    +
    7
    		)
    =
    0
  • (ii)
    (
    −
    13
    x
    −
    12
    y
    +
    31
    		)
    =
    0
  • (iii)
    (
    8
    x
    +
    24
    y
    +
    7
    		)
    =
    0
  • (iv)
    (
    −
    13
    x
    −
    10
    y
    +
    31
    		)
    =
    0
  • (v)
    (
    8
    x
    +
    22
    y
    +
    7
    		)
    =
    0
    Assignment Key
  •  1) (iv)
  •  2) (ii)
  •  3) (i)
  •  4) (i)
  •  5) (iv)
  •  6) (ii)
  •  7) (i)
  •  8) (iii)
  •  9) (iv)
  •  10) (ii)
  •  11) (v)
  •  12) (ii)
  •  13) (ii)
  •  14) (i)
  •  15) (i)
  •  16) (iv)
  •  17) (i)
  •  18) (iv)
  •  19) (i)
  •  20) (iii)
  •  21) (iii)
  •  22) (ii)
  •  23) (i)
  •  24) (iii)
  •  25) (v)
  •  26) (iv)
  •  27) (v)
  •  28) (i)
  •  29) (v)
  •  30) (v)