I/D #2: Unit O Concept 7-8: How can we derive the patterns for our Special Right Triangles?

Inquiry Activity Summary 

30, 60, 90 Degree Triangle

In the 60, 60, 60 degrees triangle we use the rule that the triangles sides all equal one, and to get a  30, 60, 90 degrees triangle we must cut the original triangle in half, leaving us with one 30 degree angle, one 90 degree angle and last one remains 60 degrees. When dividing the triangle in half, we were left with one side equaling 1/2, the side across from the 30 degrees angle. The side across from the 90 degree angle is the hypotenuse which always equals 1. The next side is the one we search for. In searching for that side we use the Pythagorean Theorem (a^2 + b^2 = c^2) to give us the accurate answer, but to do that we use a variable that represents any number and  eliminates the fraction 1/2. plugging in the variable n, and equaling any number, such as 2, the 1/2 cancels to n, and the hypotenuse to a 2n. Next we plug those numbers and variables into the Pythagorean Theorem and solve. we are left with n radical 3; giving us our last side value.

45, 45, 90 Degree Triangle 

In a  90, 90, 90, 90 degrees square all sides are equaled to one; so in order to get a 45, 45, 90 degree triangle, we must divide the square into two triangles in half, diagonally. when dividing the square, we were left with two sides still equaling to 1, while the hypotenuse is undetermined so far. So we must use the Pythagorean Theorem (a^2 + b^2 = c^2) to solve for the "r" value; leaving us with radical 2. Once again the input of the "n" only means that it can be any number, as for the value does not always equal 1.

Inquiry Activity Reflection: 

1) Something I never noticed before about special right triangles is that the values that come with the sides and angles make sense in that they are not just random values, but are calculated to make a special right triangle.

2) Being able to derive these patterns myself aids in my learning because if i get lost and somehow forget my derivatives, i can use this method of finding the values of both the angles and the sides, by either using Pythagorean Theorem or simply remembering the steps and recalling the values.