Congruence and Similarity

Lesson

Did you know that you could use similar triangles to estimate the distance from yourself to a distant object? This investigation will help you to explore how to do this.

- To reason why two triangles are similar.
- To practice problem solving with similar triangles.
- To use similar triangles in real-life situations.

- Large far away object (car, tree, house etc.)
- Paper
- Pen

- Hold your arm straight out in front of you with your thumb up.
- Shut your left eye and align your thumb with the left edge of the object.
- Without moving your thumb open your left eye and shut your right eye.
- Take note of how much your thumb has appeared to move horizontally in relation to the object.

Use the following facts and the diagram to answer the questions:

- The distance from your eyes to your thumb is approximately 10 times the distance between your eyes.
- The distance from the distant object to your thumb is 10 times the distance your thumb appears to move when you switched which eye was open.
- Both triangles are isosceles triangles

- What similarity proof can you use for the two triangles in the diagram? Why?
- Measure, or estimate the length of the distant object. Based on this information and the facts above, how far away approximately is the object from your thumb? Explain how you arrived at your answer.
- Pick two distant objects. One object should be directly behind the other and the objects should be slightly far apart from one another. Using the distance from the objects to your thumb find the distance between the two objects.

Either measure out or use a map to determine how close your estimation was.

Recognise when shapes are similar and use proportional reasoning to find an unknown length

Apply geometric reasoning in solving problems