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

Deriving the Special Square Triangles

Inquiry Activity Summary:

1. We can derive the 30-60-90 triangle from an equilateral triangle with a side length of one by using the pythagorean theorem. To create a 30-60-90 triangle, we need to form the 30 angle measure. To do this, the base can be divided equally in half. That would make the two portions 1/2 and 1/2 in side length. We already know the hypotenuse is 1 because it is an equilateral triangle. Therefore, we can find the height by using the pythagorean theorem. After doing so, the hypotenuse is found to be rad3/2. Because the base also has a denominator of 2, it would be smart to multiply all the side lengths by 2 to get fewer fractions. That would leave us with rad3 for the height (opposite to 60 degree angle measure), 1 for the base (opposite to the 30 degree angle measure) and 1/2 for the hypotenuse (opposite to the right angle). 

Multiplying by 2 does not alter their relationship because it keeps the proportions the same (it is done to all three sides). Our pattern found must include n because it would account for the change that would occur if our equilateral triangle did not have sides of 1. We keep the pattern the same because those are constants that represent the relationship between the sides.

2. We can derive the 45-45-90 triangle from a square with a side length of one by using the pythagorean theorem. To form a 45-45-90 triangle, we must cut the diagonal right corners of the square in half (in order to get the two 45 degree angle measures). That would leave us with an unknown value for the hypotenuse. We have the adjacent and opposite, we know they are 1 (it is a square). If we use the Pythagorean theorem, we find the hypotenuse is rad2. 

We must add the variable of n because the pattern constants represent the relationship between the sides when the sides are 1. To account for the change made when the sides aren't 1, we must add the n to the pattern values. The constants would remain consistent because it is being done equally.

Inquiry Activity Reflection:

1. Something I never noticed before about special right triangles is that their relationship between sides are always constant. I did not know that the "n" in their patterns stood for change in those constants and that it could equal any number as long as it stays consistent with the rest oft he triangle.

2. Being able to derive these patterns myself aids in my learning because it makes me identify and UNDERSTAND the relationship between the sides. Before, I only memorized those constants. Now I know where they come from and I know what the relationships between them mean. If I ever forget, I could derive the patterns themselves.