Similarity Write Up
The focus problem of this unit asked us to find the height of the flagpole, so that we could get a new HTHCV flag. We knew that the flag had to be a certain size, based on the height of the flagpole.
Launch
To start solving this problem, we first made an estimate on how tall we thought the flagpole was. I thought the minimum height of the flagpole was 10ft and the maximum height was 30ft. (1)We were then asked to think of a reason behind the estimates. My thought process was to compare the flagpole to the basketball hoop. I thought that the flagpole looked about three times the height of the basketball hoop, and I also assumed that the basketball hoop was 10ft tall. Similarity (2)We then looked at polygons and the concept of similarity. I quickly learned that similarity is when two polygons are the same shape but a different size. We looked at rules that help us prove similarity. We looked at rules mainly for triangles. One the rule that we used a lot was the Angle-Angle theorem. This states that if two corresponding angles on two triangles are equal than the triangles are similar. We did a lot of work on similar triangles, and at the moment I had no idea how that applied to the flagpole problem, until we looked at the shadow method. Mirror Method (4) In the mirror method, we started by looking at how the mirror method creates similar triangles, you can see that illustrated in the slideshow. The mirror on the ground creates an angle that is the same in both triangles. We also know that there is a given right angle that connects to the subjects. This makes the AA theorem true for this method. To solve for the height of the flagpole using this method, we placed the mirror down on the ground and had one of our group members step away from the mirror until she could see the top of the flagpole in the mirror. We then measured the distance from her to the mirror the distance from the mirror to the base of the flagpole, and her height. The proportion looked like this, Alexa's height = Distance from Alexa to mirror Flagpole's height Distance from flagpole to mirror We then plugged in these values (X represents the flagpole's height) 60in = 20in X 125in We cross multiplied and we found the flagpole's height to be 31.25 ft. Clinometer (5) To start the clinometer method, we looked at what makes an Isosceles triangle. Given, we know that the base angles are the same in EVERY Isosceles triangle, so we know that if one angle is 90 degrees, the other two will be 45 degrees. My group was a bit confused about how to use the clinometer at first, but it was pretty straightforward when Mr. Carter explained it to us. We measured the horizontal distance from us to the base of the flagpole, and the vertical distance from our eyes to the ground. We added these numbers to find the remaining value. The height of the flagpole we got using this method was 25.9ft. |
Shadow Method
(3)To start the shadow method we looked at how you can form two similar triangles. Using the flagpole for example, we know that there will always be a right angle in place because the shadow has to connect to the object. We also know that the sun is at the same place when we took our measurements so this creates another equal angle at the top of the triangle. These two equal angles prove that the triangles are similar (AA theorem). You can see this illustrated in the slideshow. In this method, we measured the flagpole's shadow height, and everyone in our group's shadow and height. We found our average shadow height and average height. We set up a proportion which looked like Our Height = Flagpole's height Our Shadow Flagpole's shadow When we plugged in the numbers it looked like this (X represents the unknown height of the flagpole) 64.6in = X 92.6in 475in When we first cross multiplied to solve we got X=37.25ft but when I double checked it, I got 27.6. I do not know how we got such a big number difference. Final Estimation My best final estimation for the height of the flagpole would be using the shadow method, the method that got us 27.6 feet. I think that this method is the most accurate because it clearly shows what you need to measure and the sun proves that the triangles are similar.
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