I/D2: Unit O - How can we derive the patterns for our special right triangles?

Inquiry Activity Summary


1. In order to derive a 30-60-90  triangle from an equilateral triangle, we start off with the given information that since it is an equilateral triangle, all sides are the same, and in this case they are 1. Also, since the sides are all the same, this also means that the angles are the same as well. Therefore, having a triangle add up to 180, we can conclude that each angle is 60 degrees if you divide 180 degrees by the amount of sides, which is three. To form the 30-60-90 triangle, we simply split the triangle in half. Once we do that, there is now a 90 degree angle on the interior. Since we split the triangle in half, we split the 60 degrees angle in half as well, thus making it 30 degrees on each side. We now have two 30-60-90 triangles.

 To find the sides, we know that the equilateral triangle has sides of 1. But, since we split the split the triangle in half, the side across the 30 degrees is also split in half. We know the angle across 90 degrees is still 1 because the splitting of the triangle didn't affect the side (now the hypotenuse). To find the angle across the 60 degrees with our newly made triangle, we use Pythagorean Theorem. Pythagorean theorem is a^2+b^2=c^2. The hypotenuse is c, and the angle across the 30 is a. We are looking for b, which  turns out to be radical 3 over 2. 

Lastly, we multiply each side by 2 because we want to get rid of the fractions. After multiplying by 2, our sides conclude to be a=1, b= radical 3, and c equaling to 2. The importance is that a pattern begins to form. Since a=1, we see that side c is twice of it. This creates the notion that a is equal to n, and side b is 2n. The third side concludes to be n radical 3 since 1 times radical 3 stays the same. We place n to show that there is a pattern, that the numbers from the sides don't just magically appear by memorization. There is a relationship between each sides in relation to their angles and the n shows that, and you can solve it by using n.

2. In order to derive a 45-45-90 triangle from a square, we know that a square adds up to 360 degrees and all sides are equal. We also know that a square has equal angles of 90 degrees. 

To create a 45-45-90 triangle, we draw a diagonal through the square. Because each side was 90 degrees in the square, we see that two of the angles are split because of the diagonal. Therefore, each of the angles split is 45 degrees, forming a 45-45-90 triangle. 

We know that each side from the 45 degrees remains 1 because the diagonal cut did not affect them. To find the hypotenuse, the diagonal cut, we use Pythagorean Theorem. We know that leg a is 1 and so is leg b, so we plug it in to 1^2+1^2=c^2. We then result to 2=c^2. To get our c we square root both sides and we get radical 2=c. Thus, we see that our sides is a=1, b=1, and c=radical 2.

 This is when n comes to play. We see that n is equal to 1, therefore n takes place for a and b. Since the hypotenuse is radical 2 times 1, and we know that n is equal to 1 (which both sides a and b have), we can conclude that our legs are going to be n radical 2. N represents the relationship between the sides in the 45-45-90 triangle and no matter what numbers we are given, we can find the other sides using the relationship of N. N IS POWERFUL! :) 

INQUIRY ACTIVITY REFLECTION

1. “Something I never noticed before about special right triangles is…” there is an actual reasoning behind why 30-60-90 and 45-45-90 have the sides that they do. Before I just assumed to know from memorization!
2. “Being able to derive these patterns myself aids in my learning because…” IT GOES BACK TO THE UNIT CIRCLE AND IT HELPS ME UNDERSTAND THE UNIT CIRCLE SO MUCH BETTER NOW :)