a triangle has sides 35cm,54 cm and 61 cm long fnd its aea also find the smallest of its - Maths - Heron\s Formula

Question

a triangle has sides 35cm,54 cm and 61 cm long fnd its aea also
find the smallest of its altitudes

Shridhar Kaushik · Accepted Answer

The given sides of triangle ABC are a = 35cm, b = 54cm and c = 61cm.

Semi-perimeter of triangle, S = $\frac{a + b + c}{2} = \frac{150}{2} cm$ = 75cm.

$Now, Area of triangle ABC = \sqrt{s (s - a) (s - b) (s - c)} = \sqrt{75 (75 - 35) (75 - 54) (75 - 61)} = 420 \sqrt{5} {cm}^{2} .$

Let the smallest altitude be h

We know that,

Area of triangle = $\frac{1}{2}$ × Base × height

$\Rightarrow 420 \sqrt{5} {cm}^{2} = \frac{1}{2} \times 61 cm \times h \Rightarrow h = \frac{840 \sqrt{5}}{61} cm \Rightarrow h = 13.77 \sqrt{5} cm or 30.791 cm (approx.)$

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a triangle has sides 35cm,54 cm and 61 cm long.fnd its aea. also find the smallest of its altitudes.