I/D #2: Unit O - How can we derive the patterns for our special rights triangle?

INQUIRY ACTIVITY SUMMARY:

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30-60-90 Triangle:        To begin we have to know that it is an equilateral triangle. This means that all sides and angles are equal. With that in mind we realize that the triangle has three angles, and all the angles in a triangle add up to 180 so what we do is divide 180 by 3 and get an answer of 60.
Then we cut straight down on the triangle and that gives us an angle of 30* on the top because since we divide it symmetrically the angle dives into two. If we were to add 30+30 we would get an answer of 60.  Then we created a 90* angle on the bottom left because the line created a perpendicular angle with the bottom of the triangle. ( as shown in the first picture). Since its an equilateral angle we know that all sides are equal and in this case it has a measurement of 1 on all three sides. The bottom side is 1/2 because its half of one. To find the other side (height) we use the Pythagorean Theorem ( a^2+b^2=c^2). We plug in 1/2 for a, were looking for b, and we plug in for 1 for c. This gives us an answer of rad3/2 for b. However to make the sides easier do deal with we give all sides  a variable, we will use n, and were going to give that n a value of 2n. We give it a value of two because we don't want to deal with fractions. (see picture 2) Once we have done that we get a hypotenuse of 2n a horizontal side of n, and a vertical side of n rad3. Remember n just represents any number and it keeps the relationship consistent.

    2. 45-45-90 triangle

             For the 45-45-90 triangle we begin with a square. A square has four equal sides and four 90* angles.  (see the first picture)First we drew a diagonal which  cut the 90* angle into two  pieces and that left us with  a 45 degree angle. Since all sides are equal the horizontal and the vertical sides are 1. Then we use the Pythagorean theorem to find c ( diagonal). We use 1 for a and b and that gives us an answer of rad2 (picture number 2)  Then we add n because n represents any number and it remains the relationship consistent.

INQUIRY ACTIVITY REFLECTION
       1. Something I never noticed before about right triangles is how we were able to find the ratios from the equilateral triangle and a square.
        2. Being able to derive these triangles myself aids in my learning because  I can understand how a special right triangle works and use this for concepts 7 and 8.