Question

If $\Delta JKL$ is an equilateral triangle, $JK=13x+5$ , $KL=17x-19$ , $JL=8x+35$ , how do you find $x$ and the measure of each side?

Answer 1

Hint: To solve these questions, which include finding the length of the sides of an equilateral triangle, first equate any two sides and find the value of the variable. Then substitute the variable back into each of the equations given of the side to find their lengths.

Complete step by step solution:
An equilateral triangle $\Delta JKL$ . And we have also been given the equations of its sides which are $JK=13x+5$ , $KL=17x-19$ and $JL=8x+35$

Now, we have to find the value of the variable in the given equations and then find the lengths of the sides of the given equilateral triangle.
We know that in an equilateral triangle the length of all the three sides of the triangle are the same. Therefore, we can equate the equations of any two sides and find the value of the unknown variable $x$
So, equating any two sides of the equilateral triangle, we get,
$JK=KL$
Substituting, the equations of both the sides, we get,
$\Rightarrow 13x+5=17x-19$
Now, transpose all the variable terms on the left-hand side of the equation and the constant terms on the right-hand side of the equation, to get,
$\Rightarrow 13x-17x=-19-5$
Now, simplifying the above equation by adding or subtracting the terms, we get,
$\Rightarrow -4x=-24$
Now, divide both the sides of the equation by $-4$ to get,
$\Rightarrow x=6$
Therefore, the value of the unknown variable is $x=6$ . Now, to get the lengths of the three sides of the equilateral triangle, substitute the value of $x$ in their equations to get the required answer.
For $JK\;$ :
$JK=13x+5$, substitute $x=6$ to get,
$\Rightarrow JK=13\times 6+5=83$
For $KL\;$:
$KL=17x-19$, substitute $x=6$ to get,
$\Rightarrow KL=17\times 6-19=83$
For $JL\;$:
$JL=8x+35$ , substitute $x=6$ to get,
$\Rightarrow JL=8\times 6+35=83$
Therefore, we can see that the value of $x$ is $6$ and the lengths of each of the sides of the equilateral triangle $\Delta JKL$ is $83units\;$

Note: The most common mistake that students make while solving such questions is that they forget to substitute the value of the variable back in the equations to find their values. On substituting the value of x back into the equations of sides given, each side in the triangle must yield the same length.