The Ratio That Never Changes: From Similar Triangles to Tangent
🎯 Today's mission briefing
We are learning why fixing an angle locks a right-angled triangle's side ratios forever — and how that one fact lets us measure heights we can't reach.
You'll know you've got it when:
- I can explain why two right-angled triangles with the same acute angle are similar, so their matching side ratios are equal
- I can use the shadow method (similar triangles) to find a height indirectly
- I can use the tangent ratio — opposite over adjacent — with an angle and a distance to calculate a height, and sense-check it with an estimate
The pyramid and the stick
Connects to what your guest already knows and makes them curious. Activating prior knowledge is one of the strongest predictors of new learning.
Around 2,600 years ago, a Greek traveller named Thales stood in front of the Great Pyramid and did something that reportedly stunned the Egyptian priests: he measured its height without climbing it. His entire kit was a stick and a shadow. He waited until his stick's shadow was exactly as long as the stick itself — and at that moment, he knew, the pyramid's shadow had to equal the pyramid's height. Same sun, same angle, same rule for every object in the country. Today you're going to steal his trick, then upgrade it: by the end of this lesson you'll be able to measure the tallest tree in your street from the footpath, and you'll know exactly why it works.