Question

The altitude of an equilateral triangle is \[18\] inches. What is the length of a side ?

Answer 1

Hint: As it is pretty much clear from the question that length of a side of a triangle is not given but altitude of an equilateral triangle is given. So, in order to find the length of a side, one can use the formula, \[h = \dfrac{{\sqrt 3 }}{2} \times side\], where '$h$' is the length of altitude of an equilateral triangle and 'side' is the length of a side of an equilateral triangle.

Complete step by step answer:
Formula to find the altitude of an equilateral triangle is
\[h = \dfrac{{\sqrt 3 }}{2} \times side\] ------------- (i)
Suppose, the side of an equilateral triangle is ‘a’ and the altitude of a triangle is ‘h’. As it is given the altitude (h) of the triangle is \[18\] inches, that is \[h = 18\] . So, by using the formula i.e equation (i), the side of the triangle becomes,
\[ \Rightarrow 18 = \dfrac{{\sqrt 3 }}{2} \times a\]
After simplifications (taking all the numbers on one side and variable on other side),
\[ \Rightarrow 18{\text{ }} \times {\text{ }}\dfrac{2}{{\sqrt 3 }} = a\]
Multiplying \[18 \times 2\] and keeping denominator as it is we get
\[ \Rightarrow \dfrac{{36}}{{\sqrt 3 }} = a\]

On expanding \[36\] in its prime factors the above expression can be written as
\[ \Rightarrow \dfrac{{2 \times 2 \times 3 \times 3}}{{\sqrt 3 }} = a\]
On writing \[3\] as \[\sqrt 3 \times \sqrt 3 \] we have
\[ \Rightarrow \dfrac{{2 \times 2 \times 3 \times \sqrt 3 \times \sqrt 3 }}{{\sqrt 3 }} = a\]
\[\sqrt 3 \] in the numerator and denominator cancel out as shown below
\[ \Rightarrow \dfrac{{2 \times 2 \times 3 \times \sqrt 3 \times {{\sqrt 3 }}}}{{{{\sqrt 3 }}}} = a\]
And the above expression then becomes
\[ \Rightarrow a = 2 \times 2 \times 3 \times \sqrt 3 \]
After doing multiplication the value of ‘a’ will be,
\[ \therefore a = 12\sqrt 3 \]

Hence the length of a side of the triangle is \[12\sqrt 3 \] inches.

Note: The relation between the side and altitude of the equilateral triangle is \[h = \dfrac{{\sqrt 3 }}{2} \times side\]. So, if we know the value of one of the two unknowns present in the equation, we can find the second one by shifting the terms from one side of the equation to the other using the transposition method. The final answer must be represented in simplified form after cancelling the common factors in numerator and denominator.