MathematicsIf the sides of a triangle are 25cm, 17cm and 12cm, then determine the length of the altitude on the longest side.

If the sides of a triangle are 25cm, 17cm and 12cm, then determine the length of the altitude on the longest side.


  1. A
    7.5 cm  
  2. B
    7.2 cm  
  3. C
    8.2 cm  
  4. D
    9.8 cm   

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    Solution:

    Given that,
    The three sides of the triangle are 25cm, 17cm and 12cm.
    Let the three sides and height corresponding to the longest side of the triangle are a,b,c and h  respectively.
    ⸫ a = 25cm,
    b = 17cm,
    c = 12cm.   ……. (1)
    We know, Heron’s formula for area of a triangle is given by A = s(s−a)(s−b)(s−c) ,    where s is the semi-perimeter of the triangle with sides a, b, c.
    Now, semi-perimeter (s) = a+b+c 2 .  
    ⸫ Semi-perimeter of the given triangle (s) = 25+17+12 2 ,    ⇒ s = 54 2  
    ⇒ s = 27cm.  ….. (2)
    The area of the given triangle (A)=  27(27−25)(27−17)(27−12)           [From (1) and (2)] ⇒ A = (27)(2)(10)(15)   ⇒ A= 8100 ⇒ A=  90cm 2 .    ......... (3)  We know, area of the given triangle (A) =  1 2 ×base×height.  
    ⸫ A = 1 2 ×longest side× corresponding height.   ⇒ A =  1 2 ×a×h               [⸪ on comparing, a   is the longest side with length 25 cm]
    ⇒ 90 = 1 2 ×25×h               [From (1) and (3)]
    ⇒ h= 90 12.5  
    ⇒ h = 7.2cm.
    Therefore, the height corresponding to the longest side is 7.2cm.
    Hence, option (2) is correct.
     
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