Find the area of the triangular field whose sides are \[91m,{\text{ }}98m{\text{ }}and{\text{ }}105m\]. Find the height corresponding to
the longest side.

Hint: Using Heron’s formula to find area of the triangle as three sides are known and then using basic area
formula to find height corresponding to the longest side.
Formula: $S = \dfrac{{a + b + c}}{2},\,\,\,Area = \sqrt {S\left( {S - a} \right)\left( {S - b} \right)\left( {S - c} \right)} $


Complete step by step solution:

1. As three sides of the triangle are known to us. Therefore, we use Heron’s formula to find the area of the triangle.
Here $a = 91,\,\,b = 98\,\,and\,\,c = 105$
2. Calculating \[S\](semi perimeter) by using,
$S = \dfrac{{a + b + c}}{2}$,
Substituting the values of a, b and c in the above formula,
$S = \dfrac{{91 + 98 + 105}}{2}$
$S = \dfrac{{294}}{2}$
$S = 147$
3. Heron’s Formula is given as
$Area\,of\,triangle = \sqrt {S(S - a)(S - b)(S - c)} $,
Substituting the values of \[S,{\text{ }}a,{\text{ }}b\]and\[c\].
$Area{\text{ }}of{\text{ }}triangle = \sqrt {147\left( {147 - 91} \right)\left( {147 - 98} \right)\left( {147 - 105} \right)} $
$Area{\text{ }}of{\text{ }}triangle = \sqrt {147 \times 56 \times 49 \times 42} $,
Making prime factors of numbers,
$Area{\text{ }}of{\text{ }}triangle = \sqrt {3 \times 7 \times 7 \times 7 \times 2 \times 2 \times 2 \times 7 \times 7 \times 7 \times 3 \times 2} $
$Area{\text{ }}of{\text{ }}trian\lg e = 7 \times 7 \times 7 \times 3 \times 2 \times 2$
$Area{\text{ }}of{\text{ }}triangle = 4116$
4. Therefore, the area of the triangle of given sides is $4116{m^2}$.
5. Also, we know that area of any triangle is given as $\dfrac{1}{2} \times base \times height$
6. To calculate height corresponding to the longest side. We take $105m$ as the base.
Therefore, $Area{\text{ }}of{\text{ }}triangle = \dfrac{1}{2} \times base \times height$
$Height = \dfrac{{2 \times Area{\text{ }}of{\text{ }}triangle}}{{base}}$
Taking area$ = 4116{m^2}$, base $ = 105m$ we have
$Height = \dfrac{{2 \times 4116}}{{105}}$
$Height = 78.4$
7. Hence, height to the corresponding longest side of the triangle is 78.4m.


Note:When length of three sides of a triangle are given,make sure to apply Heron's formula and do not apply the formula of the area of the triangle directly.