Concentric Equilateral Triangles

The red equilateral triangle side length 4cm sits inside the larger pink equilateral triangle such that the “border” is 1cm wide.

Screen Shot 2017-04-14 at 23.57.47

What is the ratio of the height of the red triangle to the height of the pink triangle?

Can you solve using trigonometry or only using Pythagoras?

The border is now 2cm, whilst the side length of the red triangle remains 4cm.  What is the ratio of heights now?

Screen Shot 2017-04-15 at 00.06.54

Explore what happens for other border widths.  Can you generalise for any border width w?

Geogebra file here.

Spoiler here.

I got a few solutions posted on Twitter for this, but the most elegant (so far…) has to be this from @mathforge. As he said: no Trig, no Pythagoras, just ratio.

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