I/D #2: Unit O Concept 7-8: Using the 30-60-90 triangle and using the 45-45-90 angle

30-60-90
To derive the pattern for the 30-60-90 triangles from an equilateral triangle with a side length of 1 we first need to know how an equilateral triangle is labeled. We have the information that it has the side length of 1, so since it is an EQUIlateral(meaning it has EQUAL sides) all three sides of the triangle will be one. Also, since it is equilateral triangle it means that its angles must be equal the same. This is when we absorb past knowledge about triangles. All angles of a triangle must equal to 180 when added. Therefore, in an equilateral triangle the angles will equal 60 each (60 times 3 equals 180). We can create two 30-60-90 triangle from an equilateral triangle by cutting it in half down the middle. We cut it in half down the middle first of all because we cut one of the 60 degrees angles of the equilateral into two 30 degrees, and second of all cutting a line straight down the middle will create two right angles, meaning they are 90 degrees (and now we have two 30-60-90).The lengths that the two 30 degrees angles are reflecting become 1/2 since we cut the equilateral triangle in half and its bottom side length of 1 is split to two halves. We see that the lengths reflected by the 90 degrees angles, the hypotenuse, are still.We do the Pythagorean Theorem, in relation to the special RIGHT triangles we have just formed, to get the side length of the line cutting straight down the middle. When we complete the Pythagorean Theorem we see that length is equal to radical 3 divided by 2. To get whole numbers for all our sides we multiply all the side values by 2, and the side reflected by the 60 degrees becomes radical 3, the side reflected by the 30 degrees becomes 1 and the side reflected by the 90 degrees becomes 2. We place the variable n in front of all these values to show that the pattern could be expanded. In other words, the length of 1 could be any other length, example 7, and the formula will still work.






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45-45-90
We have a square with side lengths of 1. Since it is a square, all the four side lengths are equal to 1 and all the four angles will be right angles, meaning equal to 90 degrees. We cut the square diagonally to make two special right triangles of 45-45-90. (Two 90 degrees angles in the square are split to make four 45 degrees angles).Both sides reflected by both 45 degrees angles are still  length of 1 since we did not cut the square's side lengths and instead cut it through the middle diagonally.Now, this requires us to instead find the one missing length of the diagonal line, which represents the hypotenuse in the two special right triangles we formed. Again, we utilize the Pythagorean Theorem to find this missing value. When we complete the Pythagorean Theorem we find that the value is equal to radical 2. We label all the side lengths with the variable n because we show that the value could be expanded and the lengths do not have to equal 1 in order for the rules to work.







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1. “Something I never noticed before about special right triangles is…that the 30-60-90 was derived from an equilateral triangle. Now all its measurements for its rule makes sense.
2. “Being able to derive these patterns myself aids in my learning because…I am able to apply my knowledge so that when I do not remember exactly how the rules are for special right triangles, at least now I am able to know how to find them.