Question

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. subtract the lengths of the shadows.
c. use the pythagorean theorem.
d. compare the angles of elevation.

in a proportion, if \\( \frac{6}{x}=\frac{3}{4} \\), what is the value of \\( x \\)?
a. 10
b. 8
c. 12
d. 7

a tree that is 20 feet tall casts a shadow that is 10 feet long. which proportion represents this relationship compared to a 6 - foot pole?
a. \\( \frac{20}{x}=\frac{10}{6} \\)
b. \\( \frac{6}{20}=\frac{10}{8} \\)
c. \\( \frac{20}{10}=\frac{6}{x} \\)
d. \\( \frac{10}{20}=\frac{6}{x} \\)

if triangle abc is similar to triangle def and the length of ab is 5 units while the length of de is 10 units, what is the scale factor from triangle abc to triangle def?
a. 10
b. 2
c. 5
d. \\( \frac{1}{2} \\)

Explanation:

Step1: Solve the first problem

When dealing with heights and shadows of objects, we use the concept of similar triangles. The ratio of the height of an object to the length of its shadow is the same for similar objects. So, for the building and the pole, we set up a proportion \(\frac{h}{50}=\frac{5}{2}\) (where \(h\) is the height of the building). Subtracting shadow lengths (option b) is incorrect as there's no geometric basis. The Pythagorean theorem (option c) is for right - triangle side lengths \(a^{2}+b^{2}=c^{2}\), not for this ratio problem. Comparing angles of elevation (option d) is more complex than setting up a simple proportion. So the answer is a.

Step2: Solve the second problem

Given \(\frac{6}{x}=\frac{3}{4}\), we use cross - multiplication. Cross - multiplying gives \(3x = 6\times4\). Then \(3x=24\), and dividing both sides by 3, we get \(x = 8\).

Step3: Solve the third problem

For the tree (height \(h_1 = 20\), shadow \(s_1 = 10\)) and the pole (height \(h_2 = 6\), shadow \(s_2=x\)), the proportion is \(\frac{20}{10}=\frac{6}{x}\) (height to shadow ratio). Rearranging gives \(\frac{20}{6}=\frac{10}{x}\) (equivalent proportion). Option a \(\frac{20}{x}=\frac{10}{6}\) is incorrect. Option b \(\frac{6}{20}=\frac{10}{x}\) is wrong. Option c \(\frac{20}{10}=\frac{6}{x}\) is correct. Option d \(\frac{10}{20}=\frac{6}{x}\) is incorrect.

Step4: Solve the fourth problem

The scale factor from \(\triangle ABC\) to \(\triangle DEF\) is \(\frac{DE}{AB}\) (since \(DE\) is from the second triangle and \(AB\) is from the first). Given \(AB = 5\) and \(DE = 10\), the scale factor is \(\frac{10}{5}=2\).

Answer:

1. a. Set up a proportion using similar triangles.
2. b. 8
3. c. \(\frac{20}{10}=\frac{6}{x}\)
4. b. 2