Inquiry Activity Summary:
1. 30-60-90
Step 1.) I drew the equilateral triangle and labeled all sides 1 as well as labeling the degrees as 1. Because it is an equilateral triangle, all sides are equal. If all sides are equal as each other, then it makes sense that the angles are equal to each other as well. The angles also add up to 180 degrees as well, a golden rule for triangles.
Step 2.) I splitted the triangle in two by drawing a line down the middle, creating two 30-60-90 triangles. By drawing a line down the middle, I separate the 1 value on the bottom into two 1/2 values. Also, the 60 degrees at the top becomes two 30 degree angles.
Step 3.) In order to simplify things and make it a little easier, we will be using one of the triangles for this portion of the derivation.The only thing we are adding here is "b" to the opposing side of angle 60.
Step 4.) We are going to be using good ol' classic Pythagorean Theorem to solve for "b." In step 1, we use the formula of a^2+b^2=c^2. In step 2, we plug in the values. In step 3, we square the values to get the resulting answers. We then, in step 4, subtract 1/4 to both sides to get b^2 by itself. We get an answer of b^2=3/4, but we don't want the exponent. We then square root both sides in step 5. In step 6, we get our answer as depicted below.
Step 4.) We label the "b" value we solved for on the triangle. However, we have two fractions for the triangle. We want no fractions and thus we multiply every side by two to cancel out the fractions. We then receive a hypotenuse of 2.
Step 5.) Why do we use "n"? Well, we need "n" in order to solve problems."n" allows us to find values of all sides of a 30-60-90 triangle since if we have one value of a side of a 30-60-90 triangle, we basically have all the side values. It is all a matter of solving for the other 2 hidden values. Of course, in this example we do not need to use "n" to find the other sides because they were already given. However, it is crucial to understand that without "n," it would be more difficult to identify all sides. Knowing just the "n" value is enough to completely solve a special right triangle. The sides of a 30-60-90 triangle regarding n are n, n radical 3, and 2n.
2. 45-45-90
Step 1.) I drew a beautiful square as shown below. The sides are labeled 1 and right angles shown.
Step 2.) I diagonally splitted the square to turn it in two 45-45-90 triangles. The angles are cut into two 45 degree angles and now we can derive the pattern.
Step 3.) Let us solve for one of the triangles. Ignore the radical 2=c. We are going to find "c" so we can find all sides of the triangle and consequently find the same value of "c" for the other triangle.
Step 4.) Since we know 2 of the values of the sides, we are going to be using the theorem of Pythagorean. We input the values into their proper variables, with both 1 values going into a and b, respectively. We then square the 1's to get a value of....ONE! We then add the 1's together to get 2 and square it to get radical 2. That will be our value for c.
Step 5.)We then place our glorious value of radical 2 smack daddy in the middle and we have just found the hypotenuse for TWO triangles! It's just like hitting two stones with one bird.
Step 6.) But wait, we need to put the n's in our special right triangle as well to, as said before, indicate to create a sort of formula to solve any special right triangle problems. By using n, we can easily solve for the hypotenuse or for a side. The hypotenuse is n radical 2 and the side values for n are n and n. We just hav eto multiply n by radical 2 to get n radical 2 and divide n by radical 2 to get n.
Inquiry Activity Reflection:
1. Something I never noticed before about special right triangles is that they are very easy to solve once you break up the steps to solve them one by one.
2. Being able to derive these triangles myself aids in my learning because now I can apply these lessons to my special right triangles as I now know the pattern for solving both 30-60-90 triangles and 45-45-90 triangles.
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