Question

Find the area of a triangle, two sides of which are 8 cm and 11 cm and the perimeter is 32 cm.

Answer 1

Hint: In this question, we first need to write the relation between the sides of the triangle and the perimeter given by the formula \[P=a+b+c\]. Now, on further substituting the value of perimeter and the two sides we get the third side of the triangle. Then find the area of the triangle by using the formula \[Area=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}\]. Here, s is the semi perimeter which can be calculated from \[2s=P\].

Complete step-by-step solution-
Now, let us assume the triangle as P and the sides of the triangle as a, b, c
Perimeter: Total length of the sides of a plane figure.
Area: Space covered by a plane figure.
TRIANGLE:
For any triangle having sides a, b, c then
Perimeter of the triangle is given by the formula
\[P=a+b+c\]
Area of the triangle is given by the formula
\[Area=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}\]
Here, s is called the semi perimeter of the triangle which is given by the formula
\[s=\dfrac{P}{2}\]
Now, from the given conditions we have
\[P=32,a=8,b=11\]
Now, from the formula of perimeter of a triangle we have
\[\Rightarrow P=a+b+c\]
Now, on substituting the respective values we get,
\[\Rightarrow 32=8+11+c\]
Now, on rearranging the terms we get,
\[\Rightarrow c=32-19\]
Now, on further simplification we get,
\[\therefore c=13\]
Let us now find the semi perimeter of the triangle using the respective formula
\[\Rightarrow s=\dfrac{P}{2}\]
Now, on substituting the respective value of perimeter we get,
\[\Rightarrow s=\dfrac{32}{2}\]
Now, on further simplification we get,
\[\therefore s=16\]
As we already know the formula for area of a triangle we get,
\[Area=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}\]
Let us assume the area of the triangle as A
\[\Rightarrow A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}\]
Now, on substituting the respective values in the above formula we get,
\[\Rightarrow A=\sqrt{16\left( 16-8 \right)\left( 16-11 \right)\left( 16-13 \right)}\]
Now, on further writing this in the simplified form we get,
\[\Rightarrow A=\sqrt{16\times 8\times 5\times 3}\]
Now, this can be also written as
\[\Rightarrow A=\sqrt{64\times 30}\]
Now, on further simplification we get,
\[\therefore A=8\sqrt{30}c{{m}^{2}}\]
Hence, the area of the given triangle is \[8\sqrt{30}c{{m}^{2}}\].

Note: Instead of using Heron's formula to find the area of the triangle we can also find it by multiplying the base and height and then divide by 2 which gives the area. Both methods give the same result but finding the height will be lengthy so it is better to use Heron's formula.It is important to note that while substituting the values and simplifying we need to substitute the appropriate values because there is a chance for calculation mistake which changes the final result.