where can a simple thing

like a right triangle take us? let's start with

a basic right triangle

and label its sides

a, b, and c arrange four right triangles in a square: now arrange the same four right triangles

in the same square in a different way: since the four triangles take up the same amount of space in the boxes, the squares must also take up the same amount of space.

another way to say this is the area of the red square plus the area of the blue square must equal the area of the purple square. therefore Using Pythagoras' Theorem Pythagoras' Theorem something interesting happens when we do this with an isosceles right triangle... x = 5√2 x = 4√2 try another... notice anything interesting? a pattern emerges... in any isosceles right triangle with legs y, the hypotenuse will always be y√2 i wonder if there is also a rule for right triangles

with angles of 30 and 60? or 70 and 20? or... Trigonometry because each angle has its own unique rule, if we know the angle and one side of a right triangle, we can find everything else. of course! and it leads us to a pretty cool new branch of math: what if we combine these rules with a really basic

right triangle, one with a hypotenuse of 1?

for any given angle, how long would the legs be? keeping our hypotenuse 1, as we grow and shrink our angle around the circle graph, something amazing happens... this looks crazy, but if we unravel this circle, it leads to one of the most beautiful things in math: these are still pretty far in the future, but still it's good to know the answer to the original question: EVERYWHERE the sine wave a graph can help us see this. you can think of the base leg as x and the height leg as y.

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# where can a right triangle take us?

explorations of pythagoras' theorem and where it leads. intended for a late middle/early high school audience (assumes knowledge of simplifying radicals).

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