Math, asked by leelasharma12135, 9 months ago

sides of the triangle are 13cm, 14cm and 15 cm, find the length of the
smallest altitude of the triangle.

Answers

Answered by MaheswariS
49

\textbf{Given:}

\text{Sides of the triangle are 13cm, 14cm and 15 cm}

\textbf{To find:}

\text{Smallest altitude of the triangle}

\textbf{Solution:}

\text{Let the sides be}

a=13\,cm,\;b=14\,cm,\;c=15\,cm

s=\dfrac{a+b+c}{2}

s=\dfrac{13+14+15}{2}

s=\dfrac{42}{2}=21

\textbf{Area of the triangle}

=\sqrt{s(s-a)(s-b)(s-c)}

=\sqrt{21(21-13)(21-14)(21-15)}

=\sqrt{21{\times}8{\times}7{\times}6}

=\sqrt{3{\times}7{\times}4{\times}2{\times}7{\times}2{\times}3}

=2{\times}3{\times}2{\times}7

=84\,cm^2

\textbf{Case(i):}

\text{Take, base}\;b=13\,cm

\text{But, Area=}84\,cm^2

\implies\dfrac{1}{2}{\times}b{\times}h=84

\implies\dfrac{1}{2}{\times}13{\times}h=84

\implies\,h=\dfrac{168}{13}

\implies\boxed{\,h=12.9\,cm}

\textbf{Case(ii):}

\text{Take, base}\;b=14\,cm

\text{But, Area=}84\,cm^2

\implies\dfrac{1}{2}{\times}b{\times}h=84

\implies\dfrac{1}{2}{\times}14{\times}h=84

\implies\,h=\dfrac{168}{14}

\implies\boxed{\,h=12\,cm}

\textbf{Case(iii):}

\text{Take, base}\;b=15\,cm

\text{But, Area=}84\,cm^2

\implies\dfrac{1}{2}{\times}b{\times}h=84

\implies\dfrac{1}{2}{\times}15{\times}h=84

\implies\,h=\dfrac{168}{15}

\implies\boxed{\,h=11.2\,cm}

\text{Comparing the 3 cases,}

\textbf{The smallest altitude of the triangle is 11.2 cm}

Answered by rpinfinityuniverse
6

Answer:

11.2 is the answer hope it helps u

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