I/D #2: Unit O Concept 7-8: How can we derive the patterns for special right triangles?

  The activity of how to derive the patterns for special right triangles includes a square and an equilateral triangle. Based on the angles of these shapes, you can easily define what type of special right triangle it is, whether it is a 45,45,90 triangle or a 30, 60, 90 triangle. Each side length will be equal to one. We will used what we have learned about special right triangles and relate it to different formulas (Pythagorean Theorem) or concepts (Unit Circle) that we know. 

INQUIRY ACTIVITY SUMMARY

1. 30-60-90 triangle

  This is an equilateral triangle that has a sum of 180 degrees. Since all angles and sides are equal, each angle has to be 60 degrees. To get a special right triangle out of an equilateral triangle, you will need to split the triangle in half vertically. Side A will have an angle of 30 degrees, side B with 60 degrees, and side C with 90 degrees. According to the 30-60-90 special triangle, each side is equal to n, n radical 3 and 2n accordingly. Remember, "n" is a variable. For instance, if a triangle has a side length of one more than one, the ratio for the triangles will be the same. The variable "n" tells us the the approximate length of the triangle.

  The side length of the triangle is one. Equilateral triangles all have the same length and angles. Thus, each side for this triangle is one. Since the special right triangle only occurs when the equilateral triangle is in half, then the side A will have to be equal to 1/2. As a fact, 90 degrees equals to 2n. 1/2 (which is n) multiply by 2 will give you 1 as your hypotheses length. To find side B, you can simply use the Pythagorean Theorem. Side B will then eventually equal to radical 3 over 2. Another way to solve for side B is to plug 1/2 (which is n) to n radical 3. This step will also give you the same side length for B.

2. 45-45-90 triangle

  It is given to us that the square has a side length of one. A square has all equal sides and angles. Since a square has a sum of 360 degrees, each angle is 90 degrees. To get a special right triangle out of a square, you must cut the square in half diagonally. The special right triangle will be 45-45-90. Side A and B are equal to one (a given). In a 45-45-90 triangle, it is n, n, and n radical 2 accordingly. Since one is your "n", you can simply plug in 1 to n radical 2, which will give you radical 2 for the hypotheses. 

  You can also use the Pythagorean Theorem to find side C. Side A and B are both equal to one. As you solve for c^2, you get radical 2. Radical 2 cannot be simplified any further, so radical 2 is the side length for your hypotenuse. Remember, "n" is only a variable. You could have also plug in your side A or "n" to find the length of the hypotenuse. The variable "n" is a ratio for the special right triangle whether the side length is one or more than one. Thus, it is like a constant. The variable gives the special right triangle an approximate length. 

INQUIRY ACTIVITY REFLECTION

1. SOMETHING I NEVER NOTICED BEFORE ABOUT SPECIAL RIGHT TRIANGLES IS how we can derive these triangles to fully understand why the variable "n" is so important. Once you know how to apply "n" and the Pythagorean Theorem, it becomes a lot easier to find the side length.

2. BEING ABLE TO DERIVE THESE PATTERNS MYSELF AIDS IN MY LEARNING BECAUSE I can fully understand how a special right triangle functions and use this knowledge to apply to concept 7 and 8. I can also fully recognize where these special right triangles come from. I will forever remember the variable "n" and the patterns that these triangles contributed.