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

Inquiry Activity Summary

[Property of Kelsea]

1. A 30-60-90 triangle is derived from an equilateral which is cut in half. An equilateral is a triangle where all sides are equal as well as all angles. All the angles in the equilateral triangle are 60* and all the sides are 1. When we cut the triangle in half, we get different angles and sides. half of the triangle is now made up of angles which are 90*, 60* and 30*. The 60* is the one that was not affected by the cut. The 30* is the one at the top where the angle was cut in half, which makes sense since half of 60* is 30*. The 90* comes from the bottom where the side is perfectly perpendicular to the bottom side. The sides are now different as well. The hypotenuse, the slanted on, is not affected by the cut so is remains 1. The bottom one, which is cut in half becomes 1/2. However, there is no given value for the side that has been cut. In order to find that side, we use the Pythagorean Theorem [a^2 + b^2 = c^2]. The hypotenuse is c, the bottom side is a, and w solve for b. Once we plug it all in and solve for b, the answer should be b=√3/2. Now the sides should all be as follows: a=1/2, b=√3/2, and c=1. Now, since nobody is very fond of fractions, we can multiply all the sides by 2 to eliminate the fractions. Now the sides should be a=1, b=√3, and c=2. Now we are almost done! Now we must add an n to each side number because although the angles will remain, the sides are subject to change. That means n can be any number and the angles will remain the same. So finally, one last time, lets list what the angles are: a= n, b=n√3 [n should not be behind it], and c=2n. Our 30-60-90 triangle is now derived and conquered!

[Property of Kelsea]

[Property of Kelsea]

[Property of Kelsea]

[Property of Kelsea]

2. A 45-45-90 triangle is derived when a square is cut diagonally. A square is a shape where all four sides are equal [in this case we will set them equal to 1] and all sides are 90*. When cut diagonally, the two angles change, and one side changes. Since it is cut diagonally, the two angles affected will become 45* which makes sense because half of 90 is 45. The other angle remains 90* since it was not affected by the cut and it is perfectly perpendicular to the bottom side. Two of the sides were not affected, the ones that are not slanted, so they remain 1. However, one was, the hypotenuse. In order to find the hypotenuse, we must once again use the Pythagorean Theorem [a^2 + b^2 = c^2] to find the hypotenuse. Hypotenuse will be c, the bottom side will be a, and the other side will be b. Now we plug our numbers into the equation and solve for c. The answer should b √2. Now all our angles should be as follows: a=1, b=1, c=√2. Yay! Wait ... we are forgetting something that we did in the 30-60-90 triangle. What is it? Oh Right! Our lovely buddy n. Add n to all the sides because, as stated before, the angles can remain the same but the sides are subject to change! So now, our sides should be a=n, b=n [It is 1n but since any number multiplied by 1 is that number, we do not need the 1], and c=n√2 [remember, no n behind the radical!]. Congratulations! The angles are now derived and we now know how to derive the 45-45-90 triangle! Good Job!

[Property of Kelsea]

[Property of Kelsea]

[Property of Kelsea]

Inquiry Activity Reflection

1. "Something I never noticed before about special right triangles is ..."  that the sides are found using the Pythagorean Theorem equation. [Hypotenuse for 45-45-90 and the b for the 30-60-90]
2. "Being able to derive these patterns myself aids in my learning because..." if I ever forget what the side is, I can derive it using the what I learned from this activity, which is to cut a square diagonally for a 45-45-90 or an equilateral triangle for a 30-60-90 and then use the Pythagorean Theorem to find the angle that is missing.