Question

a 12 - inch tall model casts a shadow of 4 inches. a full - scale object casts a shadow of 16 feet. how tall is the full - scale object?
46 feet
64 feet
36 feet
48 feet
a 6 - foot pole casts a 3 - foot shadow. how tall is another pole that casts a shadow 6 feet long?
18 feet
12 feet
9 feet
15 feet
a scale model of a roof has a base of 10 ft and a height of 8 ft. the actual roof has a base of 25 ft. what is the height of the actual roof?
20 ft
22 ft
18 ft
12 ft
a triangle has sides 3 feet, 4 feet, and 5 feet. another triangle has sides 6 feet, 8 feet, and 10 feet. are these triangles similar?
no, their sides are not proportional.
no, they are not congruent.
yes, they are congruent.
yes, their sides are proportional.
if two triangles are similar, what must be true about their angles?
all corresponding angles are different.
at least one angle must be equal.
all corresponding angles are congruent.
only two angles need to match.
an engineer designs two ramps inclined at the same angle. ramp a has a height of 6 ft and a length of 12 ft. ramp b has a height of 9 ft. what is the length of ramp b?
20 ft
19 ft
14 ft
18 ft
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?

Explanation:

Step1: Analyze the first problem

Set up a proportion. Let \(x\) be the height of the full - scale object. The ratio of height to shadow for the model is \(\frac{12}{4}\), and for the full - scale object is \(\frac{x}{16}\). So, \(\frac{12}{4}=\frac{x}{16}\).
Cross - multiply: \(4x = 12\times16\).
Solve for \(x\): \(x=\frac{12\times16}{4}=48\) feet.

Step2: Analyze the second problem

Let \(y\) be the height of the other pole. The ratio of height to shadow for the first pole is \(\frac{9}{3}\), and for the second pole is \(\frac{y}{6}\). So, \(\frac{9}{3}=\frac{y}{6}\).
Cross - multiply: \(3y = 9\times6\).
Solve for \(y\): \(y=\frac{9\times6}{3}=18\) feet.

Step3: Analyze the third problem

Let \(h\) be the height of the actual roof. The ratio of base to height for the model is \(\frac{10}{8}\), and for the actual roof is \(\frac{25}{h}\). So, \(\frac{10}{8}=\frac{25}{h}\).
Cross - multiply: \(10h = 25\times8\).
Solve for \(h\): \(h=\frac{25\times8}{10}=20\) ft.

Step4: Analyze the fourth problem

Check the ratios of the sides. \(\frac{3}{6}=\frac{1}{2}\), \(\frac{4}{8}=\frac{1}{2}\), \(\frac{5}{10}=\frac{1}{2}\). Since the ratios of corresponding sides are equal, the triangles are similar.

Step5: Analyze the fifth problem

By the definition of similar triangles, all corresponding angles are congruent.

Step6: Analyze the sixth problem

Let \(l\) be the length of Ramp B. The ratio of height to length for Ramp A is \(\frac{6}{12}\), and for Ramp B is \(\frac{9}{l}\). So, \(\frac{6}{12}=\frac{9}{l}\).
Cross - multiply: \(6l = 9\times12\).
Solve for \(l\): \(l=\frac{9\times12}{6}=18\) ft.

Answer:

1. d. 48 feet
2. a. 18 feet
3. a. 20 ft
4. d. Yes, their sides are proportional
5. c. All corresponding angles are congruent
6. b. 18 ft