The perimeter of a triangle is 32 cm. One side of a triangle is 9 cm longer than the smallest side and the third side is 1 cm less than 4 times the smallest side. Find the area of the triangle.


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Answer:

Area : 24 cm2

Step by Step Explanation:
  1. Let's assume the smallest side of the triangle be x cm.
  2. According to the question, one side of the triangle is 9 cm longer than the smallest side.
    The length of the side = x + 9
  3. The third side is 1 cm less than 4 times the smallest side.
    The length of the third side = 4x - 1
  4. The perimeter of the triangle is 32 cm.
    Therefore, x + (x + 9) + (4x - 1) = 32
    ⇒ x + x + 9 + 4x - 1 = 32
    ⇒ 6x = 32 + 1 - 9
    ⇒ x =  
    24
    6
     
    ⇒ x = 4
    Now, x + 9 = 4 + 9 = 13,
    4x - 1 = (4 × 4) - 1 = 15
  5. Therefore, all sides of the triangle are 4 cm, 13 cm and 15 cm.
  6. the following picture shows the triangle,

    The area of the ΔABC can be calculated using Heron's formula, since all sides of the triangle are known.
    S =  
    32
    2
      = 16 cm
    The area of the ΔABC = √S(S - AB)(S - BC)(S - CA)
    = √16(16 - 4)(16 - 13)(16 - 15)
    = 24 cm2

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