In this target you will be applying what you’ve learned in the previous three targets (A, B and C).
What information is needed to determine the height of the lamppost?
What information is needed to determine the height of the lamppost?
Example 1:
You are trying to determine the height of a tree. You are standing \(15\) feet away from the tree and your shadow is \(5\) feet. You are \(6\) feet \(6\) inches tall. How tall is the tree? Adding in the dimensions to the problem, we need to consider this problem as a dilation with a center at the edge of the shadow. Why is your height \(6.5\) ft? |
The dilation creates two similar triangles by \(AA\) similarity. It is tempting to think that the \(5\) ft length dilates to \(15\) ft. What does the length dilate to?
Example 2:
A street lamp that is \(11\) feet off the ground is shining a light on a mailbox that is \(4\)feet tall. The lamp is \(6\) feet away from the mailbox. How long is the shadow of the mailbox?
Start with this:
A street lamp that is \(11\) feet off the ground is shining a light on a mailbox that is \(4\)feet tall. The lamp is \(6\) feet away from the mailbox. How long is the shadow of the mailbox?
Start with this:
Solution:
Watch the video for the solution.
Watch the video for the solution.
Congruent Angles
In the Investigation for this target you discovered that the angle that the ball hits the side of the pool table is congruent to the angle that the ball travels away from the side after bouncing. This is because of the Angle of Incidence vs. Angle of reflection - The angle of incidence, is the angle between the incident ray and the line perpendicular to the surface at the point of incidence. The angle of reflection, is the angle reflected ray with the perpendicular line to the reflecting surface. The angle of incidence and the angle of reflection are congruent to each other.
In the Investigation for this target you discovered that the angle that the ball hits the side of the pool table is congruent to the angle that the ball travels away from the side after bouncing. This is because of the Angle of Incidence vs. Angle of reflection - The angle of incidence, is the angle between the incident ray and the line perpendicular to the surface at the point of incidence. The angle of reflection, is the angle reflected ray with the perpendicular line to the reflecting surface. The angle of incidence and the angle of reflection are congruent to each other.
Example 3:
Determine the total distance that the cue ball travels to the corner pocket.
Determine the total distance that the cue ball travels to the corner pocket.
Solution:
\(\begin{align*}
\frac{x}{2.4}&=\frac{4}{1.3}\\
\left(\frac{2.4}{1}\right)\left(\frac{x}{2.4}\right)&=\left(\frac{4}{1.3}\right)\left(\frac{2.4}{1}\right)\\
x&=\frac{9.6}{1.3}\\
x&=7.4
\end{align*}\)
Total distance = \(7.4 + 2.4\) = \(9.8\) ft
\(\begin{align*}
\frac{x}{2.4}&=\frac{4}{1.3}\\
\left(\frac{2.4}{1}\right)\left(\frac{x}{2.4}\right)&=\left(\frac{4}{1.3}\right)\left(\frac{2.4}{1}\right)\\
x&=\frac{9.6}{1.3}\\
x&=7.4
\end{align*}\)
Total distance = \(7.4 + 2.4\) = \(9.8\) ft
Quick Check
1) A building that is \(200\) feet tall casts a shadow. A person who is \(6\) feet tall and who has a shadow of \(0.75\) feet stands so that the tip of their shadow lines up with the tip of the building’s shadow. How long is the building’s shadow?
2) A boy scout group is on a hike in Starved Rock State Park to earn their Forestry badge. They need to learn about different species and sizes of trees. In order to find the height of a particular tree, they use a mirror. One of the scouts, who is \(5\) feet \(10\) inches tall, positioned himself so that he can see the top of the tree in the mirror. He placed the mirror \(16\) feet from the base of the tree and \(7\) feet away from himself. What is the height of the tree?
Quick Check Solutions
1) A building that is \(200\) feet tall casts a shadow. A person who is \(6\) feet tall and who has a shadow of \(0.75\) feet stands so that the tip of their shadow lines up with the tip of the building’s shadow. How long is the building’s shadow?
2) A boy scout group is on a hike in Starved Rock State Park to earn their Forestry badge. They need to learn about different species and sizes of trees. In order to find the height of a particular tree, they use a mirror. One of the scouts, who is \(5\) feet \(10\) inches tall, positioned himself so that he can see the top of the tree in the mirror. He placed the mirror \(16\) feet from the base of the tree and \(7\) feet away from himself. What is the height of the tree?
Quick Check Solutions