Exploring Similar Triangles and Proportional Relationships

1. A tree casts a shadow of 15 feet. At the same time, a 5-foot tall person casts a shadow of 2 feet. Are the triangles formed by the tree and its shadow similar to the triangles formed by the person and their shadow? Explain your reasoning.

• No, the shadows are too short to form triangles.

• Yes, the triangles are similar.

• Yes, but only if the person is standing on a hill.

• No, the triangles are not similar because the heights are different.

2. Two buildings are 100 feet apart. A person standing 30 feet from the first building measures the angle of elevation to the top of the second building as 45 degrees. If the first building is 60 feet tall, what is the height of the second building? Are the triangles formed similar?

• 80 feet

• 100 feet

• 130 feet

• 120 feet

3. A ramp is built to help wheelchair access. If the ramp is 8 feet long and rises 2 feet, what is the angle of elevation? If another ramp is 12 feet long, what height will it reach if it maintains the same angle? Are the triangles similar?

• Angle of elevation is approximately 14.5 degrees; height of the second ramp is 3 feet; triangles are similar.

• Angle of elevation is 20 degrees; height of the second ramp is 4 feet; triangles are not similar.

• Angle of elevation is 30 degrees; height of the second ramp is 5 feet; triangles are not similar.

• Angle of elevation is 10 degrees; height of the second ramp is 2 feet; triangles are similar.

4. In a park, two trees are planted such that the distance between them is 50 meters. If one tree is 20 meters tall and the other is 30 meters tall, can you prove that the triangles formed by the trees and the ground are similar?

• The triangles cannot be compared without knowing the angles.

• Yes, the triangles are similar.

• The triangles are similar only if the trees are the same height.

• No, the triangles are not similar.

5. A model of a building is made with a height of 3 feet and a base of 2 feet. If the actual building is 30 feet tall, what should be the base of the actual building to maintain similarity?

• 20 feet

• 10 feet

• 25 feet

• 15 feet

6. A photographer is standing 10 feet away from a statue that is 5 feet tall. If the photographer takes a picture of another statue that is 15 feet tall, how far must they stand from the second statue to keep the triangles similar?

• 20 feet

• 30 feet

• 25 feet

• 15 feet

7. A ladder leans against a wall, forming a triangle with the ground. If the ladder is 10 feet long and reaches 8 feet up the wall, what is the height of the wall if a second ladder of 15 feet is used? Are the triangles similar?

• 10 feet

• 6 feet

• 12 feet

• 14 feet

8. A flagpole casts a shadow of 12 feet. A 4-foot tall person standing next to it casts a shadow of 2 feet. Are the triangles formed by the flagpole and its shadow similar to the triangles formed by the person and their shadow?

• Yes, the triangles are similar.

• The triangles are similar by AA similarity.

• No, the triangles are not similar.

• The triangles are not similar due to different angles.

9. A triangular garden has a height of 4 meters and a base of 6 meters. If a similar garden is to be built with a height of 10 meters, what will be the length of its base?

• 18 meters

• 15 meters

• 12 meters

• 8 meters

10. A kite is flying at a height of 30 feet and casts a shadow of 40 feet. If another kite is flying at a height of 15 feet, how long will its shadow be to maintain similarity?

• 10 feet

• 25 feet

• 35 feet

• 20 feet