INQUIRY ACTIVITY SUMMARY
In this activity, we were given a square that told us to derive the pattern for 45-45-90 triangles. We were only given the side length, which was 1. We were also given an equilateral triangle and we were asked to derive the pattern for 30-60-90 triangles. The information we were given was that the side of the equilateral triangle was 1. This activity is meant for us to understand why special right triangles have these patterns.
30-60-90 Triangle
To derive a 30-60-90 Triangle, we first need to label the given information, which is that the side lengths of the equilateral triangle are 1. Then, we need to split the equilateral triangle in half. This will create two right triangles. Since the side lengths of the equilateral triangle are one, the two right triangles now have a side that is 1/2. We then need to solve the missing side using the Pythagorean Theorem. We know that one side is 1/2 and the hypotenuse is 1. We then need to square both sides. 1/2 squared is 1/4 and 1 squared is 1. Then, we need to subtract 1/4 from both sides. We will get 3/4, and finally we need to take the square root of it. The top stays as radical 3, but the bottom is 2. The final side ends up being radical 3/2. Lastly, we need to multiply the sides by two to get rid of the fractions. We will end up with radical 3, 1, and 2. A 30-60-90 Triangle has"n" in its pattern simply because "n" is a variable that represents any number. If any number were placed in to replace "n", the pattern would still work. (refer to pictures below for steps)
45-45-90 Triangle
To derive a 45-45-90 Triangle, we first need to label the given information, which is that the side lengths of the square are 1. Then, we need to split the square into two right triangles by cutting the square diagonally. We know that the lengths of both legs of the triangle are 1, so we need to use the Pythagorean Theorem to solve for the hypotenuse. We need to square 1 twice, and then add it. Since 1 squared is simply 1, 1+1 equals 2. Then, we need to take the square root of two, so the hypotenuse ends up being radical 2. The side lengths are now 1,1,radical 2. If 'n' is multiplied to this, it ends up being n, n, n radical 2. The relationship between the sides is the same, so n applies to any number because it is just a constant. " n" basically just represents a number that can be substituted in (refer to pictures below for steps).
To derive a 30-60-90 Triangle, we first need to label the given information, which is that the side lengths of the equilateral triangle are 1. Then, we need to split the equilateral triangle in half. This will create two right triangles. Since the side lengths of the equilateral triangle are one, the two right triangles now have a side that is 1/2. We then need to solve the missing side using the Pythagorean Theorem. We know that one side is 1/2 and the hypotenuse is 1. We then need to square both sides. 1/2 squared is 1/4 and 1 squared is 1. Then, we need to subtract 1/4 from both sides. We will get 3/4, and finally we need to take the square root of it. The top stays as radical 3, but the bottom is 2. The final side ends up being radical 3/2. Lastly, we need to multiply the sides by two to get rid of the fractions. We will end up with radical 3, 1, and 2. A 30-60-90 Triangle has"n" in its pattern simply because "n" is a variable that represents any number. If any number were placed in to replace "n", the pattern would still work. (refer to pictures below for steps)
45-45-90 Triangle
To derive a 45-45-90 Triangle, we first need to label the given information, which is that the side lengths of the square are 1. Then, we need to split the square into two right triangles by cutting the square diagonally. We know that the lengths of both legs of the triangle are 1, so we need to use the Pythagorean Theorem to solve for the hypotenuse. We need to square 1 twice, and then add it. Since 1 squared is simply 1, 1+1 equals 2. Then, we need to take the square root of two, so the hypotenuse ends up being radical 2. The side lengths are now 1,1,radical 2. If 'n' is multiplied to this, it ends up being n, n, n radical 2. The relationship between the sides is the same, so n applies to any number because it is just a constant. " n" basically just represents a number that can be substituted in (refer to pictures below for steps).
INQUIRY ACTIVITY REFLECTION
1. "Something I never noticed before about special right triangles is" that the Pythagorean Theorem helps derive the patterns for special right triangles.
2. "Being able to derive these problems myself aids in my learning because" now I know why these patterns are there, so I do not have to memorize things anymore and can use this logical reasoning instead to help me solve problems.
No comments:
Post a Comment