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ICSE-VIII-Mathematics

16: Understanding Shapes (Including Polygons) Class 8 Maths

with Solutions - page 5

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  • #12-i
    its each interior angle,
    Ans : each interior angle = 140Ëš
  • #12-ii
    its each exterior angle
    Ans : each exterior angle = 180Ëš - 140Ëš = 40Ëš
  • #12-iii
    the number of sides in the polygon.
    Ans : Let no. of sides = n
    ∴ 360˚/n = 40˚
    ⇒ n = 360˚/40˚ = 9
    ⇒ n = 9
    ∴ (i) 140˚ (ii) 9
  • Qstn #13
    Two alternate sides of a regular polygon, when produced, meet at the right angle. Calculate the number of sides in the polygon.
    Ans :
    Let number of sides of regular polygon = n
    AB & DC when produced meet at P such that
    ∠P = 90˚
    ∵ Interior angles are equal.
    ∴ ∠ABC = ∠BCD
    ∴ 180˚ - ∠ABC = 180˚ - ∠BCD
    ∴ ∠PBC = ∠BCP
    But ∠P = 90˚ (Given)
    ∴ ∠PBC + ∠BCP = 180˚ - 90˚ = 90˚
    ∴ ∠PBC = ∠BCP
    = ½ × 90˚
    = 45Ëš
    ∴ Each exterior angle = 45˚
    ∴ 45˚ = 360˚/n
    ⇒ n = 360˚/45˚
    ⇒ n = 8
  • Qstn #14
    In a regular pentagon ABCDE, draw a diagonal BE and then find the measure of:
  • #14-i
    ∠BAE
    Ans : Since number of sides in the pentagon = 5
    Each exterior angle = 360/5 = 72°
    ∠BAE = 180° - 72°= 108°
  • #14-ii
    ∠ABE
    Ans : In ΔABE, AB = AE
    ∴ ∠ABE = ∠AEB
    But ∠BAE + ∠ABE + ∠AEB = 180˚
    ∴ 108˚ + 2∠ABE
    = 180Ëš - 108Ëš
    = 72Ëš
    ⇒ ∠ABE = 36˚
  • #14-iii
    ∠BED
    Ans : Since ∠AED = 108˚
    [∵ each interior angle = 108˚]
    ⇒ ∠AEB = 36˚
    ⇒ ∠BED = 108˚ - 36˚
    = 72Ëš
  • Qstn #15
    The difference between the exterior angles of two regular polygons, having the sides equal to (n - 1) and (n + 1) is 9°. Find the value of n.
    Ans : We know that sum of exterior angles of a polynomial is 360°
    If sides of a regular polygon = n - 1
    Then each angle = 360Ëš/(n - 1)
    And if sides are n + 1, then each angle = 360Ëš/(n +1)
    According to the condition,
    360Ëš/(n - 1) - 360Ëš/(n + 1) = 9
    ⇒ 360˚[1/(x - 1) - 1/(x + 1)] = 9
    ⇒ 360˚ [(n + 1 - n + 1)/(n - 1)(n + 1)] = 9
    ⇒ (2 × 360)/(n2 - 1) = 9
    ⇒ n2 - 1 = (2×360)/9 = 80
    ⇒ n2 - 1 = 80
    ⇒ n2 = 1 - 80 = 0
    ⇒ n2 - 81 = 0
    ⇒ (n)2 - (9)2 = 0
    ⇒ (n + 9)(n - 9) = 0
    Either n + 9 = 0, then n = -9 which is not possible being negative,
    Or, n - 9 = 0, then n = 9
    ∴ n = 9
    ∴ No. of sides of a regular polygon = 9
  • Qstn #16
    If the difference between the exterior angle of a n sided regular polygon and an (n + 1) sided regular polygon is 12°, find the value of n.
    Ans : We know that sum of exterior angles of a polygon = 360°
    Each exterior angle of a regular polygon of 360°
    sides = 360Ëš/n
    And exterior angle of the regular polygon of (n + 1) sides = 360Ëš/(n + 1)
    ∴ 360˚/n - 360˚/(n + 1) = 12
    ⇒ 360 [1/n - 1/(n + 1)] = 12
    ⇒ 360[(n + 1 - n)/n(n + 1)] = 12
    ⇒ (30 × 1)/(n2 + n) = 12
    ⇒ 12(n2 + n) = 360˚
    ⇒ n2 + n = 36˚ (Dividing by 12)
    ⇒ n2 + n - 30 = 0
    ⇒ n2 + 6n - 5n - 30 = 0 {∵ -30 = 6 ×(-5) = 1= 6 - 5}
    ⇒ n(n + 6) - 5(n + 6) = 0
    ⇒ (n + 6)(n - 5) = 0
    Either n + 6 = 0, then n = - 6 which is not possible being negative
    Or, n - 5 = 0, then n = 5
    Hence, n = 5
  • Qstn #17
    The ratio between the number of sides of two regular polygons is 3:4 and the ratio between the sum of their interior angles is 2:3. Find the number of sides in each polygon.
    Ans : Ratio of sides of two regular polygons = 3 : 4
    Let sides of first polygon = 3n
    and sides of second polygon = 4n
    Sum of interior angles of first polygon = (2 × 3n - 4) × 90 ˚
    = (6n - 4) × 90˚ and sum of interior angle of second polygon = (2 × 4n - 4) × 90˚
    = (8n - 4) × 90˚
    ∴ ((6n - 4) × 90˚)/((8n - 4) × 90˚) = 2/3
    ⇒ (6n - 4)/(8n - 4) = 2/3
    ⇒ 18n - 12 = 16n - 8
    ⇒ 18n - 16n = -8 + 12
    ⇒ 2n = 4
    ⇒ n = 2
    ∴ No. of sides of first polygon = 3n = 3 × 2 = 6
    And no. of sides of second polygon = 4n
    = 4 × 2
    = 8
  • Qstn #18
    Three of the exterior angles of a hexagon are 40°, 51 ° and 86°. If each of the remaining exterior angles is x°, find the value of x.
    Ans : Sum of exterior angles of a hexagon = 4 x 90° = 360°
    Three angles are 40°, 51° and 86°
    Sum of three angle = 40° + 51° + 86° = 177°
    Sum of other three angles = 360° - 177° = 183°
    Each angle is x°
    3x = 183°
    ⇒ x = 183/3
    Hence x = 61
  • Qstn #19
    Calculate the number of sides of a regular polygon, if:
    Ans : Let number of sides of a regular polygon = n
  • #19-i
    its interior angle is five times its exterior angle.
    Ans : Let exterior angle = x
    Then interior angle = 5x
    x + 5x = 180°
    ⇒ 6x = 180°
    ⇒ x = 180˚/6 = 30˚
    ∴ Number of sides(n) = 360˚/30 = 12
  • #19-ii
    the ratio between its exterior angle and interior angle is 2:7.
    Ans : Ratio between exterior angle and interior angle = 2 : 7
    Let exterior angle = 2x
    Then interior angle = 7x
    ∴ 2x + 7x = 180˚
    ⇒ 9x = 180˚
    ⇒ x = 180˚/9
    = 20Ëš
    ∴ Ext. angle = 2x = 2 × 20˚ = 40˚
    ∴ No. of sides = 360˚/40 = 9
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