I/D #2: Unit O Concepts 7-8: Derive the SRTs

INQUIRY ACTIVITY SUMMARY: In class, we were each given a worksheet and: a) told to derive the pattern for the 45-45-90 triangle with a side length of one and b) told to derive the pattern for 30-60-90 triangle with a side length of one(based off of an equilateral triangle).

1. We can easily derive the 30-60-90 triangle from an equilateral triangle! The first step is to draw a vertical perpendicular line. This border will create the angles that we need to work with.
                              
    As you can see, by drawing a line, we have created a 90 degree angle and a 30 degree angle. We know that one angle is 90 degrees because it is perpendicular to the base of the triangle and we know that the other is 30 degrees as we split a previous 60 degree angle into two. Since we split the base in two, we also have to split the value. This leaves us with a base equal to half. We know that the hypotenuse is going to be equal to one because if you look at the un-seperated triangle, it is across from a 60 degree angle which gives us a side value of 1.
                          
   Here we will go on to use the Pythagorean Theorem to find the height.
                                  
   From this, we go on to make the pattern easier to work with by multiplying everything by 2, leaving us with whole numbers. We add "n" to the pattern to make it applicable to other values besides 1! 
                        
                        
2. The first step that I took was to draw a diagonal line, cutting the square into two identical triangles. From here I went on to label the degrees. Since we know that the shape was a square, that gives us an obvious 90°. Knowing this angle, we can easily label the others. The two remaining angles will be 45°. We can also go about this mathematically; as we know that a triangle must add up to 180°, we can subtract 90° and then divide it by two to equally split it. Since two of our angles are congruent, two sides of our triangle will also be congruent. We also know that the two sides are at a value of 1 as stated in the directions.
                                    
    To find the hypotenuse, we will use the Pythagorean Theorem!
                                     
    We add the variable "n" to the pattern as it allows us to apply it to any other number! As long as "n" is constant, the pattern should reflect proportional values. 
                                    

INQUIRY ACTIVITY REFLECTION:

1. “Something I never noticed before about special right triangles is…” how they came by their patterns! Before learning how to derive them, I used sole memorization to complete work.
2. "Being able to derive these patterns myself aids in my learning because.." i understand where the patterns are coming from. It also allows me to find the values should i ever forget them (this could be a lifesaver on a test)!