Question

a 9-foot pole casts a 3-foot shadow. how tall is another pole that casts a shadow 6 feet long?
a. 15 feet
b. 12 feet
c. 18 feet
d. 9 feet

a building casts a shadow 50 ft long, and a 5 ft pole casts a shadow 2 ft long. how can you calculate the height of the building?
a. set up a proportion using similar triangles.
b. use the pythagorean theorem.
c. subtract the lengths of the shadows.
d. compare the angles of elevation.

a tree and a pole cast shadows at the same time. what property allows you to use their shadows to find the tree’s height?
a. their shadows form congruent triangles.
b. the lengths of their shadows are equal.
c. both objects have the same length.
d. their shadows form similar triangles.

Explanation:

First Question:

Step1: Set up proportion

Let \( h \) be the height of the second pole. The ratio of pole height to shadow length is constant. So, \(\frac{9}{3}=\frac{h}{6}\)

Step2: Solve for \( h \)

Cross - multiply: \( 3h = 9\times6 \)
\( 3h=54 \)
Divide both sides by 3: \( h = \frac{54}{3}=18 \)

To find the height of the building, we can use the fact that the triangles formed by the building and its shadow, and the pole and its shadow are similar. So we set up a proportion using similar triangles. The Pythagorean theorem is for right - triangle side - length relationships, subtracting shadow lengths is not a valid method, and comparing angles of elevation is not the way to calculate the height here.

When a tree and a pole cast shadows at the same time, the triangles formed by the objects (tree, pole) and their respective shadows are similar. Congruent triangles would require the triangles to be identical in size, which is not the case. The lengths of their shadows are not necessarily equal, and the objects (tree and pole) do not have the same length. So the property is that their shadows form similar triangles.

Answer:

c. 18 feet

Second Question: