Question

If the perimeter of an isosceles triangle is \[11cm\]and the base is \[5cm\], its area is given by?

Answer 1

Hint: The perimeter of an isosceles triangle is given by
\[P = 2 \times (length\_of\_a\_equal\_side) + base\]
Using this formula, we will find the length of the equal side of an isosceles triangle and using this we will find the area of an isosceles triangle.
The area of a triangle when its sides are given
\[A = \sqrt {s(s - a)(s - b)(s - c)} \]
where \[s = \dfrac{{(a + b + c)}}{2}\] and \[a,b\]&\[c\]are the sides of the triangle.

Complete step by step answer:
It is given that the perimeter and base of an isosceles triangle are \[11cm\] and \[5cm\]respectively.
We know that perimeter of an isosceles triangle is
\[P = 2 \times (length\_of\_a\_equal\_side) + base\]
Let \[a\] be the length of the equal side of an isosceles triangle and \[b\] be the base of the isosceles triangle.
\[\therefore P = 2 \times (a) + b\]
Substituting the respective values to\[P\]and \[b\]we get,
\[
   \Rightarrow 11 = 2 \times (a) + 5 \\
   \Rightarrow 11 - 5 = 2a \\
   \Rightarrow 6 = 2a \\
   \Rightarrow \dfrac{6}{2} = a \\
   \Rightarrow a = 3 \\
 \]
Therefore, the length of the equal side of an isosceles triangle is \[3cm\]
Now we will find the area of an isosceles triangle using the sides of a triangle.
We have \[a = 3cm,b = 3cm\] and \[c = 5cm\]as sides of an isosceles triangle.
We know that area of a triangle when sides are given is
\[A = \sqrt {s(s - a)(s - b)(s - c)} \]
where \[s = \dfrac{{(a + b + c)}}{2}\] and \[a,b,c\]are the sides of the triangle.
Let us first calculate the values of \[s\],
\[s = \dfrac{{(3 + 3 + 5)}}{2} = \dfrac{{11}}{2} = 5.5\]
Substituting the respective values in the area formula we get,
\[A = \sqrt {5.5(5.5 - 3)(5.5 - 3)(5.5 - 5)} \]
On simplifying this we get
\[
   = \sqrt {5.5(2.5)(2.5)(0.5)} \\
   = \sqrt {17.1875} \\
   \simeq 4.14 \\
 \]
Therefore, the area of an isosceles triangle having \[P = 11cm\] and \[base = 5cm\] is \[A = 4.14c{m^2}\]which is the required answer.

Note: Isosceles triangle is a triangle which has two equal sides (i.e. \[a = b\]), so in the area of a triangle having sides we have used the same value for\[a\] & \[b\]. Also, the value of \[s\]is nothing but the half of the perimeter of the triangle that is \[s = \dfrac{P}{2}\]so we can also find it this way.