Question

lesson 2 b
reinforce understanding
indirect measurement allows you to use properties of similar polygons to find
distances or lengths that are difficult to measure directly.
for exercises 1 - 3, determine the missing height or distance.

1. a statue casts a shadow 30 feet long. at the same

time, a person who is 5 feet tall casts a shadow
that is 6 feet long. how tall is the statue?

1. a building casts a shadow 72 meters long. at the

same time, a parking meter that is 1.2 meters tall
casts a shadow that is 0.8 meter long. how tall is
the building?

1. the two triangles shown in the figure are similar.

find the distance d across red river

Explanation:

Step1: Set up the proportion for problem 1

Since the statue and the person form similar triangles (by the property of similar polygons for indirect measurement), the ratio of height to shadow length is the same. Let the height of the statue be \(h\). The proportion is \(\frac{h}{30}=\frac{5}{6}\).

Step2: Cross - multiply

Cross - multiplying gives \(6h = 30\times5\).

Step3: Solve for \(h\)

\(6h=150\), then \(h=\frac{150}{6}=25\).

Step4: Set up the proportion for problem 2

Let the height of the building be \(h\). Using the similar - triangles property (indirect measurement), the proportion is \(\frac{h}{72}=\frac{1.2}{0.8}\).

Step5: Cross - multiply

Cross - multiplying gives \(0.8h = 72\times1.2\).

Step6: Solve for \(h\)

\(0.8h = 86.4\), then \(h=\frac{86.4}{0.8}=108\).

Step7: Set up the proportion for problem 3

Let the distance across the river be \(d\). Using the similar - triangles property (indirect measurement), the proportion is \(\frac{d + 1}{18}=\frac{1}{0.9}\).

Step8: Cross - multiply

Cross - multiplying gives \(0.9(d + 1)=18\times1\).

Step9: Expand and solve for \(d\)

\(0.9d+0.9 = 18\), then \(0.9d=18 - 0.9=17.1\), and \(d=\frac{17.1}{0.9}=19\).

Answer:

1. The statue is \(25\) feet tall.
2. The building is \(108\) meters tall.
3. The distance \(d\) across Red River is \(19\) km.