Correct Answer - Option 3 : 24√5 cm
Concept:
Semi-perimeter, \(s= \frac{a+b+c}{2}\)
Now, according to Heron's formula,
Area of triangle = \(√ {s(s-a)(s-b)(s-c)}\)
Calculation:
s = (35 + 54 + 61)/2 = 75 cm
Now, Area of the triangle = \(√{75(75-35)(75-54)(75-61)}\)
\(⇒ √ {75 × 40×21×14}\)
⇒ 420√5 cm2
Let h be the longest altitude.
⇒ Area of triangle = \(\frac{1}{2}× a × h\)
⇒ 420√5 = \(\frac{1}{2}× 35 × h\)
⇒ h = 24√5 cm
Hence, the length of the longest altitude is 24√5 cm.
The length of the longest altitude is asked, i.e Value of h should be maximum.
420√5 = \(\frac{1}{2}× a × h\)
For h to be maximum, value of a should be minimum.
If we take a = 54
420√5 = 1/2 × 54 × h
h = 420√5 / 27 = 15.5√5
If we take a = 61
h = 13.7√5
So, the smaller the value of a, higher the value of h.
