SOLUTION: An equilateral triangle circumscribes a circle with radius 4. Find the height of the triangle. How was the answer 12 units? Thank you.

Question 1027679: An equilateral triangle circumscribes a circle with radius 4. Find the height of the triangle.
How was the answer 12 units? Thank you.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
each angle of an equilateral triangle is equal to 60 degrees.

if you drop a perpendicular from each of the vertices to the opposite side, you will find that intersection of these perpendiculars will be the center of the inscribed circle.

you will also find that this separates the triangle into 6 separate smaller triangles.

you will also find that each vertex of the equilateral triangle will have had their angles cut exactly in half, with each of the halves equal to 30 degrees.

the six triangles formed are all right triangles because the perpendiculars form right angles with the opposite side of the vertex from which they were formed.

all of the triangle are therefore 30,60,90 right triangles.

one leg of each of the six triangles formed is equal to the radius of the inscribed circle.

this makes one leg of each right triangle equal to 4 units.

since the opposite angle is 30 degrees, and since the sine of 30 degrees is 1/2, we can use that fact to find the length of the hypotenuse of each of those triangles.

we'll use one of those triangle to show you what happens.

in the diagram, the altitude of the equilateral triangle we chose is the line segment BD.

the triangle formed that we chose is triangle BEF.

the point E is the center of the circle and also the intersection of the 3 perpendicular we dropped from the vertices of the equilateral triangle, which is triangle ABC.

the 30 degree angles we chose is angle EBF.

the opposite side of that angle is EF which is equal to 4 units.

the hypotenuse of that triangle is EB.

we are working with triangle BEF.
the 30 degree angle of that triangle is angle EBF.
the opposite side to that 30 degree angle is side EF.
the hypotenuse of that triangle is BE.
the 90 degree angle of that triangle is angle BFE.

we know that the sine of an angle is equal to opposite / hypotenuse.

therefore sin(EBF) = EF / BE.

since angle EBF = 30 degrees and since EF = 4, we get:

sin(30) = 4 / BE.

solve for BE to get BE = 4 / sin(30) = 8.

we know that BE is part of BD.

we know that the other part of BD is ED.

we know that BE is equal to 8 and ED is equal to 4, therefore the length of BD is equal to 12.

the line segment BD is the height of the equilateral triangle ABC, therefore we have found the height of triangle ABC.

it is 12 units.

the diagram is shown below:

$$$

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