Why do this problem?
encourages learners to examine the relationships between the ratios of corresponding sides of similar right-angled triangles. The problem extends the ideas established in the problems Dotty Circle
and Where is the Dot?
using these ideas to introduce sine and cosine ratios.
The suggestions here build on learners having met the problem " Where is the Dot?". The support notes include an interactivity which makes the connection between the animation in "Where is the dot?" and the protractor
Demonstrate how the protractor works.
Ask the learners to investigate the properties of the protractor and the numbers that are produced when it is laid over the angles of each of the right-angled triangles.
What stays the same and what is different?
Working in pairs, ask them to discuss, list and justify what they notice and try to give explanations for what they find ready to share ideas with the whole group.
- Why do the numbers remain the same when the protractor is placed over either of the two acute angles?
- Why don't the numbers change when you change the radius of the protractor?
- What is the connection between the "lengths of the two adjacent sides" and the "length" of the hypotenuse?
- What relationships remain the same for any triangle? Can you explain why?
How could you use the protractor to find angles and sides of right-angled triangles with a hypotenuse that is not 1 unit long?
The interactivity below makes explicit the connections between the problem Where is the Dot?
and the protractor, by showing the lengths of the horizontal and vertical lines in a circle of unit radius.