Question

An equilateral triangle has an altitude length of 36 feet. Determine the length of a side of the triangle. ( ANSWER NEEDS TO BE IN REDUCED RADICAL FORM )

Answer 1

24√3 feet

Explanation:

Consider an equilateral triangle with sides that are 2 units in length.

If you divide an equilateral triangle vertically into two halves, you will end up with 2 smaller triangles with hypotenuse = 2 units and base = 1 unit (see attached drawing). By Pythagorean theorm, we can determine the height of the triangle to be
`√(3)` units.

This proportion of sides ( 1 : √3 : 2) is a characteristic property of a special type of right-triangle that is created by dividing an equilateral triangle into 2 halves (and is worth memorizing).

From the diagram we can see that the ratio of the dimensions of the right triangle:

base : height = 1 : √3

mathematically,

base (of right angle) / height = 1 / √3

or

base (of right angle) = (1 / √3) x height

in the question, it is given that altitude (aka) height = 36 feet

hence,

base (of right angle) = (1 / √3) x 36 feet = (36/√3) feet = 12√3 feet

base (of equilateral triangle) = 2 x base (of right angle)

= 2 x (12√3 ) = 24√3 feet

Answer 2

`L= 24√(3)\ ft`

Explanation:

Look at the attached image. By definition an equilateral triangle has 3 equal angles of 60 ° and has 3 sides of equal length.

In this case we know the length of its Height H shown in the image.

H = 36 °.

Since all angles of the equilateral triangle are 60 ° then X ° = 60 °.

So to find the L side we use the definition of the sine function:

`sin(x) = (opposite)/(hypotenuse)`

Note that in this case:

`Opposite = H=36\°\\\\Hypotenuse = L`

So:

`sin(60) = (36)/(L)`

Now we solve the equation for L

`L= (36)/(sin(60\°))`

`L= 24√(3)\ ft`