Inquiry Activity Summary:In class we were given a square with side lengths of 1 and an equilateral triangle, also with side lengths of 1. We were given directions to derive the patterns for the 45-45-90 and 30-60-90 triangles from the square and the equilateral triangle using only what we knew of the two shapes (ignoring any pre-knowledge of the 45-45-90 and 30-60-90 patterns). The following is how I accomplished this heroic task:
1). Deriving the 30-60-90 pattern with an equilateral triangle:First things first, you want to draw an equilateral triangle. We know the sides are all equal to 1 and we know that the three angles are all equal to 60 degrees in this equilateral.
altitude (top middle). Several things will change; the first and most obvious being that we will have 2 triangles. Second, the top 60 degree angle will be cut into two 30 degree angles (one for each triangle). And lastly, the bottom side of the original triangle will also split in half, resulting in two horizontal side lengths of 1/2 (again, one for each triangle). Also important is that the intersection cuts the triangle perpendicularly, meaning that you will have two 90 degree angles on each of the two new triangles. [Note: If you haven't noticed by now, we will be deriving not one 30-60-90 triangle, but two.] Take notice that we already have the 30-60-90 degree relationship established by this step.
2). Deriving the 45-45-90 pattern from a squareOkay. First I drew the square. Its sides will be one and like all squares this one will have 4 right angles.
diagonal. The results will be two right triangles, and as shown, two of the right angles will be cut in half into two 45 degree angles. We can already see the 45-45-90 degrees in out triangle, and we have two of the sides; we just need that third missing side again. But also take note that the two sides, like the two angles, are equal; this is because the sides correspond with the angle opposite to it and since they are equal angles, there will be equal sides.
Now we use the Pythagorean theorem to find that missing side. Again this is demonstrated below. We shall find that the missing side (this time "r") is equal to radical 2.
Inquiry Activity Reflection:1). Something I never noticed before about special right triangles is... that they come from a square and an equilateral triangle, and that the patterns they follow are not just numbers that happen to work every time, but they actually came from what we did in this activity.
2). Being able to derive these patterns myself aids in my learning because... learning something isn't just memorizing something with no meaning; this helps me understand what special right triangles really are.