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

2 Answers
Feb 16, 2016

Answer:

Calculated "from grass roots up"

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

Explanation:

Tony B

#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'

Feb 16, 2016

Answer:

Calculated using a quicker method: By ratio

#h= 5 1/3 sqrt(3)#
#color(red)("How is that for shorter!!!!")#

Explanation:

Tony B

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)#