Investigation 2: 30-60-90 triangles

1. Consider an equilateral triangle (all sides are equal).
   1. Explain why each of the angles measure 60°. (Hint: all the angles in a triangle always add to something.)
   2. If you fold an equilateral triangle in half (like shown in the picture above) you get two new triangles. What are the measures of the angles of these new triangles?
   3. These new triangles are better known as a "30-60-90" triangle. Why do you suppose they are called that?
      
      
2. We give special attention to the 30-60-90 triangles as well as the 45-45-90 triangles. For each of the following triangles, explain how the markings indicate that the triangles is a 30-60-90 triangle.
   

We saw that all 45-45-90 triangles are similar to each other. Likewise, all 30-60-90 triangles are similar to each other. As such, all 30-60-90 triangles are 1/2 of an equilateral triangle.

3. If ΔABC above is equilateral and BC = 2, then explain why each of the following are true.
   1. ΔBCD is a 30-60-90 triangle.
   2. AB = 2
   3. BD = 1
   4. CD = . (Hint: Pythagorean Theorem)
      
4. So our first 30-60-90 triangle really looks like
   Let's call this our "mother triangle" since all other 30-60-90 triangles are similar to it.
   
   

   Note:

   = 1.7320508075688772935274463... which never repeats or ends. We prefer to use when we describe the side length because it's a lot easier to write than 1.7320508075688772935274463... .

   
   
   1. If you enlarge the triangle above with a scale factor of 5, what will happen to each of the side lengths?
   2. Would you prefer to write or 5*1.7320508075688772935274463... ?
5. Consider this 30-60-90 triangle
   It's 1/2 the size of the last 30-60-90 triangle.
   1. How long is the short leg?
   2. Explain why the long leg is long.
      
      
6. Let's think about what happens when we take the "mother triangle" from #4 and enlarge it with a scale factor of 12. The hypotenuse would be 24 long.
1. Sketch a picture of what this new larger 30-60-90 triangle would look like.
2. How long would the short leg be?
3. How long would the long leg be (in terms of )?
   
   
   
7. Find the missing lengths in the following 30-60-90 triangles.
   
   Check your answers here.
   
   
8. Using proportionate reasoning we can find the lengths of any sides of any 30-60-90 triangle. Consider these two triangles:
   
   Both triangles are 30-60-90. The smaller triangle is our "mother triangle" and the larger one isn't scaled up by a simple number.
   Yet we can still use proportions to find the unknown lengths.
   1. Explain why the proportion is true for these two triangles.
   2. Look closely at the algebra below to see how we find the value of x.
      
      Using your calculator, what are the decimal approximations for and ?
   3. Before calculators, we would always "rationalize the denominator" like shown above in the algebra so that there would no longer be a square root in the denominator. Since we have calculators today to do the decimal approximations, we only prefer to write instead of for two reasons:
      1. It's easier to compare answers with each other if we all write our answers the same.
      2. If the 4 had been a multiple of 3 then the fraction would simplify further like in this example: .
   4. Mimic the proportion and algebra above to find the missing length in this 30-60-90 triangle.
      
      
      
9. Quick practice finding the missing lengths in the following triangles.
   (Press the small triangle to check your work and see more examples.)
   
10. Describe in your own words how to find the missing sides of a 30-60-90 triangle when you know
    1. the hypotenuse length
    2. the length of the short leg
    3. the length of the long leg