Geometry

[timothy.shaha]

How can I prove that the radius of a circle inside an equilateral triangle is one third the altitude of the triangle? Please Help Me.....

[Brent@GMATPrepNow]

A proof? What are the 5 answer choices (GMAT questions always have 5 answer choices).
Sorry, but proofs aren't part of the GMAT.

Cheers,
Brent

[Lifetron]

The radius of a circle inside an equilateral triangle is one third the altitude of the triangle

I think, it has got to do with in-radius and circum-radius. They will divide the altitude in the ratio 1:2. Not sure, which is what though.

Is the statement right ?

Atleast, we can use that for problems !

[timothy.shaha]

At least thank you for your concern. I honestly did't know that you can't
that my question is irrelevant.

[veenu08]

Property of equilateral triangle-1.All centroid, circum centre, in center meet are concurrent
2. Centroid divide the median in 2:1 ratio

In equilateral triangle length of median(altitude)- (sqrt3)/2

Using point 2, the radius of incircle will be 1/3(altitude)

Hope this clears your doubt

[Brent@GMATPrepNow]

How can I prove that the radius of a circle inside an equilateral triangle is one third the altitude of the triangle? Please Help Me.....

In case any of this comes in handy, here's the proof:

Start with this.


Draw line that bisects 60-degree angle to create 30-degree angle


Draw line from center to point of tangency to create 90-degree angle


IMPORTANT: This creates a special 30-60-90 triangle. So. let's let the radius of the circle have length 1


This means that the hypotenuse of the 30-60-90 triangle will have length 2


Finally, we can add another radius of length 1 to show that the equilateral triangle's altitude is 3 times the radius of the inscribed triangle.


Cheers,
Brent