Question

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

Answer 1

Calculated "from grass roots up"

`h= 5 1/3 xx sqrt(3)` as an 'exact value'

Explanation:

`color(brown)("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 + (a/2)^2 =a^2`...........................(1)

So we need to find `a`

We are given that the perimeter is 32 cm

So `a+a+a = 3a = 32`

So `" "a=32/3" " so " " a^2=(32/3)^2`

`a/2" " = " "1/2xx32/3" "=" "32/6`

`(a/2)^2= (32/6)^2`

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

Substituting these value into equation (1) gives

`h^2+(a/2)^2 =a^2" "-> " "h^2+(32/6)^2=(32/3)^2`

`h= sqrt( (32/3)^2-(32/6)^2)`

There is a very well known algebra method hear where if we have
`(a^2-b^2) = (a-b)(a+b)`

also `32/3= 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 -> 2xx4^2`
`96-> 2^2xx2^2xx3xx2`

giving:

`h= sqrt( 1/6^2xx2^2xx2^2 xx2^2xx4^2xx3)`

`h= 1/6xx2xx2xx2xx4xxsqrt(3)`

`h=32/6 sqrt(3)`

`h= 5 1/3 xx sqrt(3)` as an 'exact value'

Answer 2

Calculated using a quicker method: By ratio

`h= 5 1/3 sqrt(3)`
`color(red)("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:

`32/3 =10 2/3`

So `1/2` of one side is `5 1/3`

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

`(10 2/3)/2 = h/(sqrt(3))`

so `h= (1/2 xx 10 2/3) xx sqrt(3)`

`h= 5 1/3 sqrt(3)`