1) A light post casts a shadow on a small boy. The boy, who is \(4\) feet \(6\) inches tall, notices that his shadow is \(3\) feet \(4\) inches long. He decides that if he measures how far he is from the light post, he can determine how tall the post is from the ground. Explain how he can figure out the height of the light post. Use mathematical reasoning.
2) Using the information from problem #1, if the boy is standing exactly \(5\) feet from the light post, how tall is the post. Show all reasoning.
3) Maria accidentally drops her mirror on the ground in front of her. Before picking it up she notices that she can see the reflection of sprinkler on the ceiling. Maria thinks that she can determine how high up the sprinkler is attached. Explain how Maria can do this. Use mathematical reasoning.
4) If Maria's eyes are \(5\) ft from the floor and she is standing \(4\) ft from the dropped mirror and the mirror is 7.75 feet from the point directly below the sprinkler (as shown in the picture). Determine the height of the sprinkler. |
6) The sun casts its shadow on a tree and, nearby the sun also casts on a boy. Both the tree and the boy and their shadows create similar triangles. Explain why they are similar.
7) The boy, Gerald, understands that these similar triangle are formed. So, he measures both the shadow of the tree and his own shadow. Knowing his own height, he can figure out the height of the tree? Explain how he can do this.
7) The boy, Gerald, understands that these similar triangle are formed. So, he measures both the shadow of the tree and his own shadow. Knowing his own height, he can figure out the height of the tree? Explain how he can do this.
9) A player hits a pool ball off bottom rail \(2\) feet from the left rail and \(1.5\) from the \(2\) feet from the bottom rail as shown in the picture. The ball travels an additional \(7.5\) feet into the pocket. Explain why similar triangles are formed. What is the length and width of the pool table?
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11) A laser is shot from point "laser" into a mirror. Where should you place the laser (ordered pair) so that you hit the target?
12) A laser is shot from point "laser" into a mirror. Where should you place the laser (ordered pair) so that you hit the target?
13) A laser is shot from point "laser" into a mirror. Where should you place the laser (ordered pair) so that you hit the target?
14) A laser is shot from point "laser" into a mirror. Where should you place the laser (ordered pair) so that you hit the target?
15) To determine the height of a Chicago business building, you place a mirror on the ground and stand where you can see the top of the building in the mirror. You are \(64\) inches tall (to your eyes), the mirror is \(20\) feet from the base of the building, and you are standing \(40\) inches from the mirror.
a) Draw a figure that represents this situation (with names of vertices).
b) Write a similarity statement based on the two triangles formed.
c) Calculate the height of the building.
16) A student incorrectly set up and solved the problem #15 as follows. Explain the error.
\(\begin{align*}\\
\dfrac{64}{240} &= \dfrac{40}{x}\\
\left(\dfrac{40}{40}\right)\left(\dfrac{64}{240}\right) &= \left(\dfrac{40}{x}\right)\left(\dfrac{64}{64}\right)\\
\dfrac{2560}{9600} &= \dfrac{2560}{64x}\\
9600 &= 64x
\end{align*}\)
\(x = 150\) inches or \(12.5\) feet
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