Question

The sides of a triangle are \[5y,6y\] and \[9y\]. Find an expression for its area \[A\].
1) \[10{y^2}\sqrt 2 \]
2) \[5{y^2}\sqrt 2 \]
3) \[20{y^2}\sqrt 2 \]
4) \[100{y^2}\sqrt 2 \]

Answer 1

Hint: As we know that the three sides of the non-equilateral triangle are given and we need to find the expression for the area of the triangle. So, we will use the heron’s formula approach to find the expression. As we know the formula of heron’s we will apply it directly by finding the value of \[s\], which is given by \[s = \dfrac{{a + b + c}}{2}\] where \[a,b\] and \[c\] are three sides of the triangle. Now, we will use formula of area of triangle (using heron’s formula approach) is given by \[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \].

Complete step-by-step answer:

First let the given three sides \[5y,6y\] and \[9y\] of the triangle as \[a,b\] and \[c\] respectively.
As we know that, the sides are not the exact values but the values have variable \[y\] in it so we will use the heron’s formula to find the expression for the area of a triangle.
Thus, the formula for the area of triangle is \[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \].
Here, we have the values of \[a,b\] and \[c\], we just have to find the value of \[s\] which is known as the semi-perimeter.
So, the value of \[s\] can be determined using the formula \[s = \dfrac{{a + b + c}}{2}\] where \[a,b\] and \[c\] are three sides of the triangle.
Thus, substitute the values of the triangle given and we get,
\[
  s = \dfrac{{5y + 6y + 9y}}{2} \\
   = \dfrac{{20y}}{2} \\
   = 10y \\
 \]
Hence, the value of \[s\] is given as \[10y\] units.
Further, we will substitute the values of \[s\] and the values of \[a,b\] and \[c\] in the formula of area of triangle which is given by \[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \].
Thus, we get,
\[
   \Rightarrow A = \sqrt {10y\left( {10y - 5y} \right)\left( {10y - 6y} \right)\left( {10y - 9y} \right)} \\
   \Rightarrow A = \sqrt {10y\left( {5y} \right)\left( {4y} \right)\left( y \right)} \\
   \Rightarrow A = \sqrt {15{y^2}\left( {4{y^2}} \right)} \\
   \Rightarrow A = \sqrt {200{y^4}} \\
\]
Hence, we have the value of the area of the triangle.
As we can see it can be simplified further by taking the square root of the value.
The value of \[\sqrt {200} \] is given by \[\sqrt {200} = 10\sqrt 2 \] and the value of \[\sqrt {{y^4}} \] is given by \[\sqrt {{y^4}} = {y^2}\].
Thus, substitute the values in the expression obtained for the area of the triangle.
We get,
\[A = 10\sqrt 2 {y^2}\] or \[A = 10{y^2}\sqrt 2 \]
Hence, the expression for the area of the triangle is \[A = 10{y^2}\sqrt 2 \].
Thus, option A is correct.

Note: Remember the heron’s formula to find the area of the triangle. It is necessary to find the value of \[s\] which is commonly known as the semi-perimeter of the triangle. Semi perimeter is the perimeter when divided by 2 or the half of the perimeter. Heron’s formula is used when only the sides of the triangle are given and height is not as just use the sides without having the value of height and find the area.