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ICSE-VIII-Mathematics

20: Area of Trapezium and a Polygon Class 8 Maths

with Solutions - page 5

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  • Qstn #3
    The two parallel sides and the distance between them are in the ratio 3 : 4 : 2. If the area of the trapezium is 175 cm2, find its height.
    Ans : Let the two parallel sides and the distance between them be 3x, 4x, 2x cm respectively
    Area = 1/2 (sum of parallel sides) x (distance between parallel sides)
    = 1/2 (3x + 4x) × 2x = 175 (given)
    ⇒ 7x × x = 175
    ⇒ 7x2 = 175
    ⇒ x2 = 25
    ⇒ x = 5
    Height i.e. distance between parallel sides = 2x = 2× 5 = 10 cm
  • Qstn #4
    A parallelogram has sides of 15 cm and 12 cm; if the distance between the 15 cm sides is 6 cm; find the distance between 12 cm sides.
    Ans : Base, AB = 15 cm
    Distance between 15 cm sides
    i.e., height DP = 6 cm
    ∴ Area of ∥gm = Base × Height
    = AB × DP
    = 15 × 6
    = 90 cm2
    Let BQ be distance 12 cm sides
    ∴ AD × BQ = area of ∥gm ABCD
    ∴ 12 × BQ = 90
    BQ = 90/12
    BQ = 15/2
    = 7.5 cm
  • Qstn #5
    A parallelogram has sides of 20 cm and 30 cm. If the distance between its shorter sides is 15 cm; find the distance between the longer sides.
    Ans : Let ABCD be the ∥gm in which BC = 30 cm and CD = 20 cm
    Distance between shorter sides,
    i.e., CQ = 15 cm
    ∴ area of ∥gm = AB × CQ
    = 20 × 15
    = 300 cm2
    Again BC × AP = Area of ∥gm
    30 × AP = 300
    ⇒ AP = 300/30
    ⇒ AP = 10 cm
    ∴ Distance between larger sides is = 10 cm
  • Qstn #6
    The adjacent sides of a parallelogram are 21 cm and 28 cm. If its one diagonal is 35 cm; find the area of the parallelogram.
    Ans :
    First we find area of ΔABC,
    Sides are, a = 28 cm
    B = 35 cm
    And c = 21 cm
    S = (a + b + c)/2
    = (28 + 35 + 21)/2
    = 84/2
    = 42 cm
    ∵ Diagonal of ∥gm divides it into two equal parts.
    ∴ area of ∥gm = 2 × area of ΔABC
    = 2 × 294
    = 588 cm2
  • Qstn #7
    The diagonals of a rhombus are 18 cm and 24 cm. Find:
  • #7-i
    its area ;
    Ans : Diagonal of rhombus are 18 cm and 24 cm.
    area of rhombus = 1/2 ×Product of diagonals
    = 1/2 × 18 × 24
    = 216 cm2
  • #7-ii
    length of its sides.
    Ans : Diagonals of rhombus bisect each other at right angles.

    ∴ OA = ½ × 24 = 12 cm OB =1/2 × 18 = 9 cm
    In right ∠d ΔAOB
    ∴ Side of rhombus = 15 cm
  • #7-iii
    its perimeter;
    Ans : Perimeter of rhombus = 4 × side
    = 4 × 15
    = 60 cm
    (i) 216 cm2(ii) 15 cm(iii) 60 cm
  • Qstn #8
    The perimeter of a rhombus is 40 cm. If one diagonal is 16 cm; find :
  • #8-i
    its another diagonal
    Ans : Perimeter of rhombus = 40 cm
    side = ¼ ×40 = 10 cm
    One diagonal = 16 cm
    Diagonals of rhombus bisect each other at right angles.


    ∴ Diagonal BD = 6 × 2 = 12 cm
  • #8-ii
    area
    Ans : Area of rhombus = ½ × product of diagonals
    = ½ × 12 × 16
    = 96 cm2
    ∴ (i) 12 cm(ii) 96 cm2
  • Qstn #9
    Each side of a rhombus is 18 cm. If the distance between two parallel sides is 12 cm, find its area.
    Ans : Each side of the rhombus = 18 cm
    base of the rhombus = 18 cm
    Distance between two parallel sides = 12 cm
    Height = 12 cm
    Area of the rhombus = base x height = 18× 12 = 216 cm2
  • Qstn #10
    The length of the diagonals of a rhombus is in the ratio 4 : 3. If its area is 384 cm2, find its side.
    Ans : Let the lengths of the diagonals of rhombus are 4x, 3x.
    ∴ Area of the rhombus = ½ (Product of its diagonals)
    = ½ (4x × 3x)
    = 384 (given)
    ⇒ 6x2 = 384
    ⇒ x2 = 64
    ⇒ x = 8 cm
    ∴ Diagonals are 4 × 8 = 32 cm and 3(8) = 24 cm
    ∴ OC = 16 cm and OD = 12 cm

    Hence, side of the rhombus = 20 cm
  • Qstn #11
    A thin metal iron-sheet is rhombus in shape, with each side 10 m. If one of its diagonals is 16 m, find the cost of painting its both sides at the rate of ₹ 6 per m2.
    Also, find the distance between the opposite sides of this rhombus.
    Ans : Side of rhombus shaped iron sheet = 10 m and one diagonals (AC) = 16 m
    Join BD diagonal which bisects AC at O
    The diagonals of a rhombus bisect each other at right angle
    ∴ AO = OC = 16/2 = 8 m
    Now in right ΔAOB
    AB2 = AO2 + BO2
    ⇒ (10)2 = (8)2 + BO2
    ⇒ 100 = 64 + BO2
    ⇒ BO2 = 100 - 64
    = 36
    = (6)2
    ∴ BO = 6 m
    ∴ BD = 2 × BO
    = 2 × 6
    = 12 m
    Now, area of rhombus = (d1 × d2)/2
    = (16 × 12)/2
    = 96 m2
    Rate of pairing = ₹ 6 per m2
    ∴ Total cost of painting both sides = 2 × 96 × 6
    = ₹ 1152
    Distance between two opposite sides = Area/Base
    = 96/10
    = 9.6 m
  • Qstn #12
    The area of a trapezium is 279 sq.cm and the distance between its two parallel sides is 18 cm. If one of its parallel sides is longer than the other side by 5 cm, find the lengths of its parallel sides.
    Ans : Area of trapezium = 279 sq.cm
    Distance between two parallel lines (h) = 18 cm
    ∴ Sum of parallel sides = (Area × 2)/Height
    = (279 × 2)/18
    = 31 m
    Let shorter side, CD = x
    Then longer side = x + 5
    ∴ x + x + 5 = 31
    ⇒ 2x = 31 - 5 = 26
    ⇒ x = 26/2 = 13
    ∴ Shorter side = 13 cm
    And longer side = 13 + 5
    = 18 cm
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