# The perimeter of an equilateral triangle is 32 centimeters. How do you find the length of an altitude of the triangle?

Feb 16, 2016

Calculated "from grass roots up"

$h = 5 \frac{1}{3} \times \sqrt{3}$ as an 'exact value'

#### Explanation: $\textcolor{b r o w n}{\text{By using fractions when able you do not introduce error}}$color(brown)("and some times things just cancel out or simplify!!!"

Using Pythagoras

${h}^{2} + {\left(\frac{a}{2}\right)}^{2} = {a}^{2}$...........................(1)

So we need to find $a$

We are given that the perimeter is 32 cm

So $a + a + a = 3 a = 32$

So $\text{ "a=32/3" " so " } {a}^{2} = {\left(\frac{32}{3}\right)}^{2}$

$\frac{a}{2} \text{ " = " "1/2xx32/3" "=" } \frac{32}{6}$

${\left(\frac{a}{2}\right)}^{2} = {\left(\frac{32}{6}\right)}^{2}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Substituting these value into equation (1) gives

${h}^{2} + {\left(\frac{a}{2}\right)}^{2} = {a}^{2} \text{ "-> " } {h}^{2} + {\left(\frac{32}{6}\right)}^{2} = {\left(\frac{32}{3}\right)}^{2}$

$h = \sqrt{{\left(\frac{32}{3}\right)}^{2} - {\left(\frac{32}{6}\right)}^{2}}$

There is a very well known algebra method hear where if we have
$\left({a}^{2} - {b}^{2}\right) = \left(a - b\right) \left(a + b\right)$

also $\frac{32}{3} = \frac{64}{6}$ so we have

h= sqrt( (64/6-32/6)(64/6+32/6)

h= sqrt( (32/6)(96/6)

h= sqrt( 1/6^2xx32xx96

By looking at the 'factor tree' we have
$32 \to 2 \times {4}^{2}$
$96 \to {2}^{2} \times {2}^{2} \times 3 \times 2$

giving:

$h = \sqrt{\frac{1}{6} ^ 2 \times {2}^{2} \times {2}^{2} \times {2}^{2} \times {4}^{2} \times 3}$

$h = \frac{1}{6} \times 2 \times 2 \times 2 \times 4 \times \sqrt{3}$

$h = \frac{32}{6} \sqrt{3}$

$h = 5 \frac{1}{3} \times \sqrt{3}$ as an 'exact value'

Feb 16, 2016

Calculated using a quicker method: By ratio

$h = 5 \frac{1}{3} \sqrt{3}$
$\textcolor{red}{\text{How is that for shorter!!!!}}$

#### Explanation: If you had an equilateral triangle of side length 2 then you would have the condition in the above diagram.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We know that the perimeter in the question is 32 cm. So each side is of length:

$\frac{32}{3} = 10 \frac{2}{3}$

So $\frac{1}{2}$ of one side is $5 \frac{1}{3}$

So by ratio, using the values in this diagram to those in my other solution we have:

$\frac{10 \frac{2}{3}}{2} = \frac{h}{\sqrt{3}}$

so $h = \left(\frac{1}{2} \times 10 \frac{2}{3}\right) \times \sqrt{3}$

$h = 5 \frac{1}{3} \sqrt{3}$