Inquiry Activity Series:
1. 30-60-90 triangle
In order to derive this triangle, we first need to begin with an equilateral triangle and go from there. The equilateral triangle's characteristic is that it has the same side measures for each of the three sides. In this case, our equilateral triangle will have side measures of 1. Another thing we should take note of is that an equilateral triangle is also an equiangular triangle which means all of the angles are of the same measure as well. So, because a triangle's angles must always add up to a sum of 180, we can concur that each of the three angles will be 60 degrees (180/3=60).
Now that we remember the basic knowledge of our equilateral angle, we can begin to shape it into a 30-60-90 triangle. The first thing we want to do is split the triangle in half. By doing this, we create a 90 degree angle inside. We also, by dividing the triangle in half, we divide an angle in half. By splitting this angle we create a 30 degree angle on one side of the equilateral triangle. Thus, we technically create two 30-60-90 degree triangles--but we only want to look at one.
Above, we can see the transformation from an equilateral triangle to a 30-60-90 triangle. We have the angles, so now we want to find the actual sides--the important part. As you can see in the photo, 1/2 is written across the 30 degrees. I was able to get the value 1/2 with simple logic. Each side length was 1 and because we split the triangle in half, I also split that bottom side in half. So we know that the side across the 30 degrees is 1/2 and the side across the right angle is 1. Now the question is, what is the side across the 60 degrees, the side we created by splitting the triangle in half. We are going to determine this side with the help of our friend Pythagoras, and his Pythagorean Theorem.
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Pythagorean theorem: let legs be "a" and "b" and hypotenuse be "c"
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In the above picture, I used the basic knowledge of the Pythagorean Theorem to find our 2nd leg. After working it out (look at the picture), I discovered that our unknown side has the length of
√3/2. Finally, we know all sides of the 30-60-90 triangle!
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| I placed the letters that the values corresponded with in the Pythagorean Theorem. |
We a
re not completely done quite yet. Dealing with the fractions can be quite annoying at times (correction: all the time) so we want to multiply each side by 2 in order to create nice numbers. Side "a" would be 1, side "b" would be √3, and side "c" would be 2. If you notice, even if we did not multiply by 2, side "c" is twice the amount of "a" and side "b" is basically side "a" multiplied by √3. This is where we begin to derive the 30-60-90 triangle. The relationships I pointed out are found in multiples of these sides. If we add those "n" into the picture, we see this relationship. Our side "a" (across the 30 degrees) would be "n" because it is only 1. Side "b" (across 60 degrees) would be n√3 because it is side "a" multiplied by √3 (as we discussed). Side "c" (across the 90 degrees) would be 2n because our hypotenuse is twice the length of side "a". The reason why we place "n" is to emphasize the fact that this relationship is always there with these angles even with different numbers from the ones we found in the above picture. It is like ratios where they do not all have the same numbers but have the same relationship (ex: 1:2 is the same as 2:4).
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| The final 30-60-90 triangle derived from the equilateral triangle. |
2. 45-45-90 Triangle
We want to derive this triangle from a square. Refreshing our knowledge of squares, we know that all sides are equal, all angles add up to 360, and each angle is equal to each other. Using quick mental math we know that each angle will have to be 90 degrees (360/4=90). Our square we will be using has side lengths of 1. From here, we want to create a 45-45-90 triangle and the only way to do that is draw a diagonal through the square. Like we saw in the 30-60-90 triangle, we are going to see our angles split. Two angle will be split in half (because of the diagonal) so those angles will be 45 degrees. We now have a 45-45-90 triangle.
Now, we have both sides across the 45 degrees: they are equal to 1. But now, we want to know what the measure of the hypotenuse (the dotted line) is in order to find the relationship between these numbers. Let us return to the Pythagorean Theorem!
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| Knowing the two legs allows us to find our hypotenuse, which is radical 2. |
We now know each of our sides. Both legs are equal to 1 while the hypotenuse, we just found, is equal to
√2. Comparing the sides with one another, we see that the legs are 1 while the hypotenuse is radical 2 multiplied by the legs (1). These are the values when we use a square with the sides of 1. However, the idea we want to understand is that this relationship extends beyond one specific triangle with specific lengths. This is where we substitute "n" into the triangle. We want "n" to take the place of 1 (because we have ONE "n"). So our legs (a&b) are going to be "n". Our hypotenuse (across the 90 degrees), because it is 1 times radical 2 (radical 2 times the length of side "a" or "b"), will be n times radical 2. "N" allows the possibility of different lengths and measures of these sides but still contains the relationship found in the 45-45-90 triangle.
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| We found the relationship between the sides of a 45-45-90 triangle! |
Inquiry Activity Reflection:
“Something I never noticed before about special right triangles is…” is that they are derived from these sort of shapes (the equilateral triangle and square) in order to justify why we have these numbers (they are not just randomly given).
“Being able to derive these patterns myself aids in my learning because…” if I forget the values of the sides, I can easily use my knowledge of pythagorean theorem to find them.
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