Question

Find the area of an isosceles triangle whose base is 16 cm and length of each of the equal sides is 10 cm.

Answer 1

Hint: Use the Heron’s formula to find the area of the triangle which is given by the formula \[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \] , where \[s = \dfrac{{a + b + c}}{2}\]
In this question the base of the triangle is given and also other two sides are given since isosceles triangle has two sides equal, so by using those lengths of the side of the triangle we will find the semi perimeter of the triangle and then substituting this in Heron’s formula we will find the area of an isosceles triangle.

Complete step-by-step answer:
Let the sides of the triangle whose area is to be calculated be \[a,b,c\]
So we can write
The length of the first side of triangle \[a = 16\;cm\]
The length of the second side of triangle \[b = 10\;cm\]
The length of the third side of triangle as triangle is isosceles triangle is \[c = 10\;cm\]
Now as we know semi-perimeter of a triangle is given by the formula \[s = \dfrac{{a + b + c}}{2}\] , so by substituting the value of the length of the side of triangle we get
 \[
  s = \dfrac{{16 + 10 + 10}}{2} \\
   = \dfrac{{36}}{2} \\
   = 18\;cm \\
 \]
Hence we get semi perimeter \[s = 18cm\]
Now we substitute the value of semi perimeter \[s\] in Heron’s formula to find the area of the isosceles triangle, hence we can write
 \[
  A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \\
   = \sqrt {18\left( {18 - 16} \right)\left( {18 - 10} \right)\left( {18 - 10} \right)} \\
   = \sqrt {18 \times 2 \times 8 \times 8} \\
   = \sqrt {2304} \\
   = 48\;c{m^2} \\
 \]
Hence the area of the isosceles triangle is \[ = 48\;c{m^2}\]
So, the correct answer is “ \[ = 48\;c{m^2}\] ”.

Note: Heron’s formula to find the area of the triangle requires the length of all sides of the triangle which is one of the demerits of this formula so to us this formula we need to find all the sides of the triangle but this formula can give the area of the triangle without the measurements of the angles of the triangle.