Thread: Triangle inscribed in a Hexagon

1. Triangle inscribed in a Hexagon

Hey sorry if this is in the wrong place and if the picture is too small but I need some help

What can you say about a triangle that is inscribed in a hexagon, where the vertices of the triangle hits the midpoint of 3 sides of a hexagon?
if there is something about it please explain it to me so I can understand better, thanks.

2. Re: Triangle inscribed in a Hexagon

I don't see a picture...

3. Re: Triangle inscribed in a Hexagon

sorry i just added it right now

4. Re: Triangle inscribed in a Hexagon

Is it a regular hexagon or an arbitrary hexagon?

5. Re: Triangle inscribed in a Hexagon

it is a regular hexagon

6. Re: Triangle inscribed in a Hexagon

ABCDEF is aregular hexagon. Name the three midpoints that form the triangle

7. Re: Triangle inscribed in a Hexagon

Hello, gfbrd!

What can you say about a triangle that is inscribed in a regular hexagon,
where the vertices of the triangle are the midpoint of 3 sides of a hexagon?

What do they want me to say?
. . It has three sides.
. . It has three angles.
. . It is equilateral.
. . It has 60o angles.

Okay, I'll get serious . . .

A regular hexagon is composed of six equilateral triangles of side a.
Consider the upper half of the hexagon.
Code:
            : - - a - - :
*  *  *  *  *
* .         . *
a *   .       .   * a
*=================*
*       .   .       *
*         . .         *
*  *  *  *  *  *  *  *  *
: - - a - - : - - a - - :
We have an isosceles trapezoid.
The side of the triangle is the median.
Its length is the average of the lengths
. . of the two parallel sides.
Hence, the side of the triangle is $\tfrac{3}{2}a.$

The triangle's perimeter is $\tfrac{3}{4}$ of the hexagon's perimeter.
The triangle's area is $\tfrac{3}{8}$ of the hexagon's area.

8. Re: Triangle inscribed in a Hexagon

There are two additional triangles which can be drawn meeting the general requirements

9. Re: Triangle inscribed in a Hexagon

Great thanks for your help everyone.

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area of largest triangle in a hexagon

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