The red equilateral triangle side length 4cm sits inside the larger pink equilateral triangle such that the “border” is 1cm wide.
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?
Explore what happens for other border widths. Can you generalise for any border width w?
Geogebra file 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.
Another one from the fabulous Don Steward:
You could of course just go straight for the algebraic proof but it does require a level of confidence with surds. So you might want to scaffold this task. Maybe start by putting some numbers in for the radius of the smaller semi-circle, maybe 2. You could then do it again with 4 and ask students if they are convinced by that. (Here are some examples to warn against the dangers of extrapolating what appears to be a pattern). If you do take the numbers approach it’s good calculator practice. Can you type the whole expression for the area of the curved shape into the calculator to get an exact answer?
And here is a little GeoGebra drawing to go with it.