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

Inquiry Activity Summary

For the 45-45-90 triangle, we take a square and cut it in half diagonally. We cut diagonally in order to make two of the sides 45 degrees (cutting 90 in half). We can get the hypotenuse by using the pythagor theorem and solving for c. We use the variable 'n' in order to show that this is a ratio, and can be expanded or condensed.

1) In order to get a 30-60-90 triangle from an equilateral triangle, we have to cut it down the middle.This makes the hypotenuse equal to 1 and the bottom side equal to 1/2, using the pythagorean theorem we get the last side which is radical 3 over 2. The final step is to multiply all our sides by 2 so that we get 2, 1, and radical 3. In order to show that this is a ratio and can be used with larger or smaller numbers, we use the variable n.

http://www.freemathhelp.com/images/lessons/triangle306090-3.gif

2) In order to derive a 45-45-90 triangle from a square, we have to cut it in half diagonally, making two of the angles 45 (because we cut 90 in half). This gives us a hypotenuse of radical 2 ( from the original square side) And the other two side are both 1. If we use the pythagorean theorem, we can check this and it is true.

http://hotmath.com/hotmath_help/topics/45-45-90-triangles/triangle1.gif

3) It is important to note that we use variables in both of these to show that they can maintain the same ratio even if the numbers are altered, they all still have the same relationship.

Inquiry Activity Reflection

1) Something I never noticed before about special right triangles is how they both come from other shapes, such as the square and the equilateral triangle, and we just have to cut them in half.

2) Being able to derive these patterns myself aids in my learning because if I ever forget the relationship that these triangles have, I can always go back and solve it from memory.