Sam is walking in a straight line towards a lamp post which is $8 \mathrm{~m}$ high. When he is $12 \mathrm{~m}$ away from the lamp post, his shadow is $4 \mathrm{~m}$ in length. When he is $8 \mathrm{~m}$ from the lamp post, what is the length of his shadow?
(A) $1 \frac{1}{2} \mathrm{~m}$
(B) $2 \mathrm{~m}$
(C) $2 \frac{1}{2} \mathrm{~m}$
(D) $2 \frac{2}{3} \mathrm{~m}$
(E) $3 \mathrm{~m}$
Sam is walking in a straight line towards a lamp post which is $8 \mathrm{~m}$ high. When he is $12 \mathrm{~m}$ away from the lamp post, his shadow is $4 \mathrm{~m}$ in length. When he is $8 \mathrm{~m}$ from the lamp post, what is the length of his shadow?
(A) $1 \frac{1}{2} \mathrm{~m}$
(B) $2 \mathrm{~m}$
(C) $2 \frac{1}{2} \mathrm{~m}$
(D) $2 \frac{2}{3} \mathrm{~m}$
(E) $3 \mathrm{~m}$

## Solution

As Sam approaches the lamp post, we can visualize his position, as shown.

Since $\triangle A B C$ and $\triangle A D E$ are similar, the lengths of their corresponding sides are proportional. To determine Sam's height $\mathrm{h}$, we solve $\frac{\mathrm{h}}{4}=\frac{8}{16}$, and therefore $\mathrm{h}=2 \mathrm{~m}$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-423.jpg?height=352&width=572&top_left_y=941&top_left_x=1226)

As Sam moves to a position that is $8 \mathrm{~m}$ from the lamp post we now have the situation, as shown.

Using similar triangles as before, we can now calculate, $\mathrm{L}$, the length of the shadow.

Thus, $\frac{\mathrm{L}}{2}=\frac{\mathrm{L}+8}{8}$.

Using the property of equivalent fractions, $\frac{\mathrm{L}}{2}=\frac{4 \mathrm{~L}}{8}=\frac{\mathrm{L}+8}{8}$.

Thus, $4 \mathrm{~L}=\mathrm{L}+8$

$$
\begin{aligned}
& 3 \mathrm{~L}=8 \\
& \mathrm{~L}=2 \frac{2}{3} \mathrm{~m}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-423.jpg?height=347&width=458&top_left_y=1355&top_left_x=1278)