Question

Sandy is trying to measure the height of a nearby flagpole using a mirror as shown in the diagram. The mirror is 6 meters away from the flagpole and 2 meters away from Sandy. The height to her eyes is 157 centimeters, from which she can clearly see the top of the flagpole. How many centimeters tall is the flagpole?
A.169
B.471
C.89
D.456

Answer 1

To solve for the height of the flagpole, we can use similar triangles.

Let's denote:
- \( h \) as the height of the flagpole (in centimeters),
- \( h_s = 157 \) cm as the height of Sandy's eyes,
- The distance from Sandy to the mirror is \( d_s = 2 \) m,
- The distance from the mirror to the flagpole is \( d_f = 6 \) m.

We can start by converting all distances to the same units. Since we will ultimately be accounting for heights in centimeters, we'll convert the distances into centimeters:
- \( d_s = 2 \) m = 200 cm,
- \( d_f = 6 \) m = 600 cm.

Now, we set up the ratio of the height of the flagpole (\(h\)) to the distance from the mirror to the flagpole (\(d_f = 600\) cm) and the height of Sandy's eyes (\(h_s = 157\) cm) to the distance from Sandy to the mirror (\(d_s = 200\) cm).

Using the property of similar triangles:

\[
\frac{h}{600} = \frac{157}{200}
\]

Now, cross-multiply to solve for \( h \):

\[
h \cdot 200 = 157 \cdot 600
\]

Calculating \( 157 \cdot 600 \):

\[
157 \cdot 600 = 94200
\]

So, we have:

\[
200h = 94200
\]

Now, divide both sides by 200 to find \( h \):

\[
h = \frac{94200}{200} = 471
\]

Thus, the height of the flagpole is \( 471 \) cm.

The correct answer is **B. 471**.