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

INQUIRY ACTIVITY SUMMARY:
1) 30º 60º 90º TRIANGLE

  The 30-60-90 triangle comes from an equilateral triangle, which has equilateral angles of 60 degrees. The equilateral triangle is cut in half, making: 2 triangles, (2) 30º degree angles, and (2) 90º degree angles.  Then we cut it in half and (b) will equal to 1/2 because its half of 1, while the other untouched side (c) the hypotenuse would remain as 1. However, one side remains missing with its value (a), and in order to find it we use the  Pythagorean Theorem (a^2 + b^2 = c^2).

As you can see from the pic above to the left we use it and it should 1/2 squared should be 1/4 and now we subtract it from 1 and it should be radical 3/4 but 4 is a perfect square so we just have a equaling radical 3/2 so now we know that (a) is radical 3/2 (b) is 1/2 and (c) is 1. So now were not done because we dont want a fraction so all we have to do is multiply by 2 on each side and by looking at the pic above on the right side you see that when we multiply by 2 on (a) it should now be radical 3 and (b) should be 1 lastly (c) should be 2. Then we use use "n" to represent any value for the triangle creating n radical 3 on (a) 1n or just n on(b) and 2n on (c). 

2) 45º 45º 90º TRIANGLE

The 45-45-90 triangle comes from cutting a square directly in half diagonally. Cutting the square diagonally in half will create: 2 triangles, (4) 45º degree angles, and (2) 90º degree angles. We just have to focus on one since we know that its a 45º 45º 90º triangle we know that (a) and (b) have to be the same but by looking at the pic above we know that they are both 1. To look for (c) we need to use the Pythagorean Theorem (a^2 + b^2 = c^2).

To find (c) we do 1 squared plus 1 squared equals c squared. (c) equals radical 2. Then we add "n" to represent any value for the triangle to create n radical 2 that is (c) and n or 1n for (a) and n or in as well for (b) because remember its a 45º 45º 90º triangle and you know that (a) and (b) have to be the same.

INQUIRY ACTIVITY REFLECTION:

  “Something I never noticed before about special right triangles is…” how they got those numbers as in the sides like radical 2 and what the n had to do with all of these in the first place. How similar it is for deriving a special right triangle is.

“Being able to derive these patterns myself aids in my learning because…” it gives me a better understanding on hoe they got the numbers and what n really meant instead instead of just knowing that it just had to be there and why someone chose these numbers for a reason and if i forget i can always remember where these patterns were derived from.