Question

Find the area of a triangle two sides of which are $ 18cm $ and $ 10cm $ and the perimeter is $ 42cm $

Answer 1

Hint: First we have to define what the terms we need to solve the problem are.
Perimeter is the total length of the sides in a given two-dimensional or a three-dimensional shape.
An area of a triangle is the total region that is enclosed by three sides of any triangle.
Formula used:
The formula for the area of the triangle is $ \sqrt {s(s - a)(s - b)(s - c)} $ here s is the semi perimeter which means half of the given perimeter. And \[a,b,c\] are the sides of the given triangle.

Complete step by step answer:
First, we see the given know values which are area of a triangle two sides of which are $ 18cm $ and $ 10cm $ and the perimeter is $ 42cm $ which means $ a = 18cm $ and $ b = 10cm $ and still c is unknown yet
Also, the semi perimeter formula is the half the given perimeter that is $ \dfrac{{perimeter}}{2} $
Thus the perimeter given is $ 42 $ and hence the semi perimeter value is $ \dfrac{{perimeter}}{2} $ = $ \dfrac{{42}}{2} $ = $ 21cm $
Thus, we found the semi perimeter of the given problem which is $ s = 21cm $
But since the only unknown value is c, applying formula to find c which is perimeter minus the sides of the given triangle that means, $ c = 42 - (18 + 10)cm $
Solving this we get the value of $ c = 14cm $ so now all the values are know \[a,b,c\]and s too
Therefore: Area of triangle = $ \sqrt {s(s - a)(s - b)(s - c)} $ = $ \sqrt {21(21 - 18)(21 - 10)(21 - 14)} $ (all the values in multiplication inside the square root)
Hence further following $ \sqrt {21(21 - 18)(21 - 10)(21 - 14)} $ = $ \sqrt {3 \times 7 \times 3 \times 7 \times 11} c{m^2} $ since three and seven are common terms take it out of square root and multiply then $ \sqrt {3 \times 7 \times 3 \times 7 \times 11} c{m^2} = 21\sqrt {11} c{m^2} $
Hence area of triangle which is required is $ 21\sqrt {11} c{m^2} $

Note: Since perimeter is given as 42cm but in the area of triangle formula we need to substitute half of the perimeter which is s and hence we divide into two and then we found the s, afterwards we applied it in the area of triangle formula.