# : The perimeter of an isosceles triangle is 90 m. If the length of the two equal sides is $\dfrac{3}{4}$ of the length of the unequal side, find the dimensions of the triangle.

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**Hint:**With the given perimeter ,we can form a linear equation with two variables and by solving it we get the dimensions of the triangle.

**Complete step-by-step answer:**Step 1:

It is given that it is an isosceles triangle. In an isosceles triangle any two sides are equal .

We are given the perimeter of the triangle and we know that the perimeter of the triangle is just the sum of its three sides.

Step 2 :

Let the length of the sides be x , y , z.

Perimeter = x+y+z

Since it is an isosceles triangle, x = y

Therefore , perimeter = x + x +z

$ \Rightarrow 90 = 2x + z$………….(1)

Step 3:

It is given that the length of the two equal sides is $\dfrac{3}{4}$ of the length of the unequal side.

$ \Rightarrow x = \dfrac{3}{4}z$

Cross multiplying we get

$

\Rightarrow 4x = 3z \\

\Rightarrow \dfrac{4}{3}x = z \\

$

Now lets substitute the above value in equation (1)

$ \Rightarrow 90 = 2x + \dfrac{4}{3}x$

Taking LCM , we get

$ \Rightarrow 90 = \dfrac{{6x + 4x}}{3}$

Cross multiplying we get,

$

\Rightarrow 270 = 10x \\

\Rightarrow \dfrac{{270}}{{10}} = x \\

\Rightarrow x = 27m \\

$

Now we have obtained the value of x and as it is an isosceles triangle y = 27 m

Step 4 :

To find the value of z , substitute the value of x in equation (1)

$

\Rightarrow 90 = 2(27) + z \\

\Rightarrow 90 = 54 + z \\

\Rightarrow 90 - 54 = z \\

\Rightarrow z = 36m \\

$

Therefore the value of z=36m

**Therefore, the dimensions of the isosceles triangle are 27 m , 27 m , 36 m.**

**Note:**In this problem we have used a substitution method to solve the linear equation but we can also use elimination method to solve them.