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

Answer:

$L= 24\sqrt{3}\ ft$

Step-by-step 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:

$sin(x) = \frac{opposite}{hypotenuse}$

Note that in this case:

$Opposite = H=36\°\\\\Hypotenuse = L$

So:

$sin(60) = \frac{36}{L}$

Now we solve the equation for L

$L= \frac{36}{sin(60\°)}$

$L= 24\sqrt{3}\ ft$

Answer 2

Answer:

24√3 feet

Step-by-step 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 $\sqrt{3}$ 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