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

Inquiry Summary Activity

1. 30-60-90

   The given triangle is an equilateral triangle and all of its angles are 60 degrees and each side is equal to 1. To get the 30-60-90 triangle you must draw a line down the center of the triangle, leaving one angle being 90 degrees, the other one is 30 degrees, and the other angle is 60 degrees. Like the equilateral triangle these all add up to have a total of 180 degrees. Since the hypotenuse of the triangle remains as 1, we get 1/2 for one of the sides-the x value-since the line was drawn down the center of the triangle and the original number for that side was 1. With this information we use the Pythagorean theorem to find the missing side value. When you solve for the missing side value-the y value- you end up with y^2 equaling radical 3/4 but since the 4 can break down into 2x2 the answer is radical 3/2. Next you multiply each side by 2. When you multiply the angle opposite of 30 degrees you should end up with "n" after multiplying 1/2 by 2, when you multiply the hypotenuse by two you should get "2n" since the value of the hypotenuse was 1, and for the other side you multiply 2 by radical 3/2, the two's cancel out and you're left with radical 3. The "n" values are there to show that the value can keep expanding. 

2. 45-45-90

   We are given a square with all equal sides whose values are all 1. In order for us to find our 45-45-90 triangle we first draw a line that goes from one corner of the square to another. The 90 degree angle is the corner that does not have a line drawn on it and the other two angles are both 45 degrees. Since we have our x and y values for the triangle we use the Pythagorean theorem to solve for the hypotenuse. The hypotenuse ends up being radical 2 and the other two sides remain as 1. Since the value can keep expanding we use "n", thus giving us the x and y sides being "n" and the hypotenuse ends up as "n radical 2." 

Inquiry Activity Reflection

Something I never noticed before about special right triangles is that "n" can expand. 

Being able to derive these patterns myself aids in my learning because I now know why each value is what it is.