Tim, our school director, would like to buy us a flag for our school. There are certain regulations for how big a flag can be in comparison to it's height, but no one knows how tall our flagpole is! Using different similarity methods, find out the height of the HTHCV flagpole.
Starting the Problem
In the beginning, we went outside and estimated how tall the flagpole was. By estimating how many me-s I would have to stack up in order to get to the top, I thought the flagpole would be about 25-30 feet tall (or 5-6 Iza-s).
Our next step was to venture into similarity and what that means. Our class definition was:
Corresponding angles are equal, corresponding sides are proportional. or Same shape, different sides.
An example of two similar shapes would be a square that has side lengths of 2, and a square that has side lengths of 4. This is a 1:2 scale, meaning that the second square is twice as large as the first.
Before Solving
Before we were able to use different methods to solve the problem, we had to understand some basic principles about triangles. This is shown in my work below.
In these worksheets, we proved and disproved different theorems about triangles. To review, these are the ways that you can prove similarity about triangles:
Angle Angle theorem (AA) Side Side Side theorem (SSS) Side Angle Side theorem (SAS)
These will be helpful when proving our methods of calculating the height of the flagpole.
Shadow Method
Because the sun's light comes down at a particular angle, we have shadows. Our shadows are proportional to our own height, and likewise for other objects. When standing up, we create a 90 degree angle with the ground, so do other objects. This here shows that a triangle formed by us and our shadow is similar to the triangle formed by the flagpole and it's shadow at the same time of day and location because of the AA theorem. We used this method to set up a proportion that allowed us to solve for the flagpole's height algebraically: our shadow/our height = flagpole shadow/flagpole height (unknown)
Mirror Method
The mirror method was our second way of solving this problem. First, we set a mirror down a little ways away from the flagpole. Then, we took steps back away from the mirror until we could see the tip of the top in the mirror. This creates 2 equal angles that prove similarity by the AA theorem: The 90 degree angle created by us and the floor = the 90 degree angle created by the flagpole and the floor, and the angle created by the reflection of light coming into and out of the mirror. By measuring the distance between us and the mirror, the distance between the mirror and the object, and our height, we could create the following proportion to solve for the height of the flagpole: our height/us to mirror = flagpole height (unknown)/flagpole to mirror
Clinometer Method
A clinometer is an object that allows a person to measure what angle they are looking at. A person using this tool will look through a straw attached to the flat edge of a protractor. A weighted string will hang down showing what angle the person is looking up at. When we stood up, we created a 90 degree angle with the ground, We then took steps back from the flagpole until we were looking up at a 45 degree angle.
We know that a triangle's angles will always add up to 180 degrees, and 180-(90+45)=45, so now we know all angles of our triangle that goes from our eyes, across to the flagpole, up to the top of the flag pole, and then connecting back to our eyes. An isosceles triangle is a triangle with 2 equal sides and when a triangle has 2 equal sides, it also has 2 equal angles. So because our triangle has two 45 degree angles, we know that the two lines coming out of the 90 degree angle have to be equal, otherwise the angles would change with the length change.
With this information, we can now measure the distance from us to the flagpole and get the height from our eyes up to the top. This number added to our height will give us the height of the flagpole.
Final Estimate
In this process, we ended up with 3 different answers for the height of the flagpole:
Shadow: 24'10" Mirror: 18' Clinometer: 24'6"
Because the mirror method seems to most easy to mess up in my opinion (depending on how much of the top you consider to be enough that you can see, it could change the angle pretty drastically), I would cancel that out as an outlier. With that, I'd take the average between the other two and get either 24'8", or just assume that the company that made it sold it with a label of 25'.
Problem Evaluation
Something I really loved about this unit was that I got to practice my algebra, especially my cross multiplication. Teachers and students have been trying to explain the concept of cross multiplication to me since the fourth grade and this was the first time I ever understood it. Now I understand that it just has to do with common denominators.
I also really liked that we did a deep refresher on basic geometry so that now I can take rules and theorems and apply them. Look how in depth my proofs were!
Self Evaluation
I think I did a great job on this unit. I went into depth on all of my work and I ended up getting the material really well. I not only can use this information to solve the height of the flagpole, but I showed that I was able to use it to figure out more complex problems. When I took a placement test recently at Southwestern college, a lot of this unit was very useful. In addition, I showed mastery of the subject in my test and rubric questions.