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Sides of a triangle are in the ratio \[12:17:25\] and its perimeter is \[540\,{\text{cm}}\] . Find its area.

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Hint: Write down the given parameters. Recall the process of how to find the value of quantities when their ratios are given. Use the formula of perimeter to find the sides of the triangles. Then use these values to calculate the value of the area of the triangle.

Complete step-by-step answer:
Given, ratios of the sides as \[12:17:25\]
Perimeter of the triangle, \[P = 540\,{\text{cm}}\]
Let us first find the sides of the triangle.
Let ABC be the triangle, with the ratios of the sides as
 \[{\text{AB}}:{\text{BC}}:{\text{CA}} = 12:17:25\]
Let \[x\] be a constant such that \[{\text{AB}}:{\text{BC}}:{\text{CA}} = 12x:17x:25x\]
Perimeter of a triangle is given by the sum of its sides, so perimeter of triangle ABC will be,
 \[P = {\text{AB}} + {\text{BC}} + {\text{CA}}\]
Putting the values of \[P\] and the sides AB, BC and AC we get
 \[
  540\, = 12x + 17x + 25x \\
   \Rightarrow 540 = 54x \\
   \Rightarrow x = 10
\]
Therefore, the sides are
 \[{\text{AB}} = 12x = 12 \times 10 = 120\,{\text{cm}}\]
 \[{\text{BC}} = 17x = 17 \times 10 = 170\,{\text{cm}}\]
 \[{\text{AC}} = 25x = 25 \times 10 = 250\,{\text{cm}}\]
Area of a triangle is given by,
 \[{\text{Area}} = \sqrt {s(s - a)(s - b)(s - c)} \] (i)
where \[s\] is the semi perimeter and \[a\] , \[b\] and \[c\] are its sides.
Here,
 \[
  s = \dfrac{P}{2} \\
   \Rightarrow s = \dfrac{{540}}{2} \\
   \Rightarrow s = 270{\text{cm}}
\]
And \[a\] , \[b\] and \[c\] are
 \[a = {\text{AB}} = 120{\text{cm}}\]
 \[b = {\text{BC}} = 170{\text{cm}}\]
 \[c = {\text{AC}} = 250{\text{cm}}\]
Putting these values of \[s\] , \[a\] , \[b\] and \[c\] in equation (i), we get
 \[
  {\text{Area}} = \sqrt {270(270 - 120)(270 - 170)(270 - 250)} \\
   \Rightarrow {\text{Area}} = \sqrt {270 \times 150 \times 100 \times 20}
\]
 \[
   \Rightarrow {\text{Area}} = \sqrt {81000000} \\
   \Rightarrow {\text{Area}} = 9000{\text{c}}{{\text{m}}^{\text{3}}}
\]
Thus, the area of the triangle is \[9000{\text{c}}{{\text{m}}^{\text{3}}}\] .
So, the correct answer is “\[9000{\text{c}}{{\text{m}}^{\text{3}}}\] ”.

Note: When the values are given in ratios always check for the condition in the question using which we can write an equation adding or subtracting the quantities given in ratios, in this way we can find the values of the quantities. Area of a triangle can also be written in the form, \[{\text{area}} = \dfrac{1}{2} \times {\text{base}} \times {\text{height}}\] but use this formula only when the height of the triangle is given or when we can calculate it.