856. The area of a regular hexagon of side 2√3 cm is :
     

1. 1. 18√3 cm²
      

   
   
   

   2. 12√3 cm²
      

   
   
   

   3. 36√3 cm²
      

   
   
   

   4. 27√3 cm²

   
   
   

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1. View Hint View Answer Discuss in Forum

   Area of regular hexagon =	3√3	× (side)²
   	2

   
   

   =	3√3	× 2√3× 2√3
   	2

   
   = 18√3 cm²

   Correct Option: A

   Area of regular hexagon =	3√3	× (side)²
   	2

   
   

   =	3√3	× 2√3× 2√3
   	2

   
   = 18√3 cm²

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857. The ratio of length of each equal side and the third side of an isosceles triangle is 3 : 4. If the area of the triangle is 18√5 square units, the third side is
     

1. 1. 16 units
      

   
   
   

   2. 5√10 units
      

   
   
   

   3. 8√2 units
      

   
   
   

   4. 12 units

   
   
   

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1. View Hint View Answer Discuss in Forum

   Using Rule 1 and 3,
   Let Sides = 3x, 3x and 4x Semi perimeter
   

   =	3x + 3x + 4x	= 5x
   	2

   
   ∴ ∆ = √5x(5x - 3x)(5x - 3x)(5x - 4x)
   = √5x × 2x × 2x × x
   ∴ 2√5x² = 18√5
   ⇒ x² = 9 ⇒ x = 3
   ∴ Third side = 4x = 4 × 3 = 12 units

   Correct Option: D

   Using Rule 1 and 3,
   Let Sides = 3x, 3x and 4x Semi perimeter
   

   =	3x + 3x + 4x	= 5x
   	2

   
   ∴ ∆ = √5x(5x - 3x)(5x - 3x)(5x - 4x)
   = √5x × 2x × 2x × x
   ∴ 2√5x² = 18√5
   ⇒ x² = 9 ⇒ x = 3
   ∴ Third side = 4x = 4 × 3 = 12 units

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858. The ratio of sides of a triangle is 3 : 4 : 5. If area of the triangle is 72 square unit, then the length of the smallest side is :
     

1. 1. 4√3 unit
      

   
   
   

   2. 5√ unit
      

   
   
   

   3. 6√ unit
      

   
   
   

   4. 3√ unit

   
   
   

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1. View Hint View Answer Discuss in Forum

   Using Rule 1,
   Here, (3x)² + (4x)² = (5x)²
   ∴ It is a right angled triangle.
   So, Area of the triangle
   

   =	1	× 3x × 4x = 6x²
   	2

   
   ∴ 6x² = 72 ⇒ x² = 12
   ⇒ x = 2√3
   Hence, Smallest side = 3x = 6√3 units

   Correct Option: C

   Using Rule 1,
   Here, (3x)² + (4x)² = (5x)²
   ∴ It is a right angled triangle.
   So, Area of the triangle
   

   =	1	× 3x × 4x = 6x²
   	2

   
   ∴ 6x² = 72 ⇒ x² = 12
   ⇒ x = 2√3
   Hence, Smallest side = 3x = 6√3 units

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859. If the length of each side of an equilateral triangle is increased by 2 unit, the area is found to be increased by 3 + √3 square unit. The length of each side of the triangle is
     

1. 1. √3 unit
      

   
   
   

   2. 3 unit
      

   
   
   

   3. 3√3 unit
      

   
   
   

   4. 1 + 3√3 unit

   
   
   

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1. View Hint View Answer Discuss in Forum

   Using Rule 6,
   Side of equilateral triangle = x units.
   

   ∴	√3	[(x + 2)² - x²] = 3 + √3
   	4

   
   

   ∴	√3	(4x + 4) = 3 + √3
   	4

   
   ⇒ √3x + √3 = 3 + √3 = √3x = 3
   ⇒ x = 3 units

   Correct Option: A

   Using Rule 6,
   Side of equilateral triangle = x units.
   

   ∴	√3	[(x + 2)² - x²] = 3 + √3
   	4

   
   

   ∴	√3	(4x + 4) = 3 + √3
   	4

   
   ⇒ √3x + √3 = 3 + √3 = √3x = 3
   ⇒ x = 3 units

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860. What is the area of the triangle whose sides are 9cm, 10cm and 11cm?
     

1. 1. 30 cm²
      

   
   
   

   2. 60cm²
      

   
   
   

   3. 30√2 cm²
      

   
   
   

   4. 60√2 cm²

   
   
   

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1. View Hint View Answer Discuss in Forum

   Using Rule 2 and 3,
   Semi-perimeter,
   

   S =	9 + 10 + 11	= 15 cm
   	2

   
   Area of triangle
   = √s(s - a)(s - b)(s - c)
   = √15(15 - 9)(15 - 10)(15 - 11)
   = √15 × 6 × 5 × 4 = 30√2 cm²

   Correct Option: C

   Using Rule 2 and 3,
   Semi-perimeter,
   

   S =	9 + 10 + 11	= 15 cm
   	2

   
   Area of triangle
   = √s(s - a)(s - b)(s - c)
   = √15(15 - 9)(15 - 10)(15 - 11)
   = √15 × 6 × 5 × 4 = 30√2 cm²

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