I/D #2: Unit O: How can we derive the patterns for our special right triangles?

Inquiry Activity Summary:

For this activity we had to derive the special right triangles not from the unit circle though. We had to derive the two special kinds ,45-45-90 and 30-60-90 right triangles. Each are completely different we had to find out what n was and why n couldn't just be a number. 

How can we derive the 45-45-90 triangle from an square with a side length of 1? 

We know that if we are given a square with side lengths of 1 we will have two right triangles if we split the square in half from the corners. For a 45-45-90 triangle we know the base and height are the same,1. If we do the Pythagorean theorem we will see that the hypotenuse will equal radical 2. N is there to be any given value, without n the triangle sides cannot be altered to match the initial constants, it is multiplied by the initial constants. For example if n were to equal 2 then the base and height are 2. The hypotenuse will equal 2 radical 2. 

How can we derive the 30-60-90 triangle from an equilateral triangle with a side length of 1?

To derive the pattern for the 30-60-90 triangles we have to cut an equilateral triangle in half straight down the middle. Each side length of the triangle is one so when we split the triangle in half the base turns into 1/2. We also split the triangle in  half to get 30 degrees as one of the angles. Then we will have a 30-60-90 triangle. Then we notice that we do not have the height of the triangle, we have to use the Pythagorean Theorem. Once you get the constants they can be altered to get rid of the ugly fractions by multiplying by two ( look at picture). The constants are the same because they are all proportional and everything was multiplied equally. 

Inquiry Activity Reflection: 

 Something I never noticed before about special right triangles is how we have to tweak equilateral triangles and squares to get the sides for n. 

Being able to derive these patterns myself aids in my learning because now I can refer to this square or triangle if I ever forget what the sides of the triangle equal.