Nicely similar
If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
Problem
Nicely Similar printable sheet
In the diagram below, a right-angled triangle has a line descending from its right angle, perpendicular to the hypotenuse.
If the hypotenuse length is 100cm, and if the line splits the base into 36cm and 64cm parts, what are the side lengths of the original right-angled triangle?
For another way to see this arrangement of right-angled triangles you may like to look at the problem called a Matter of Scale .
Getting Started
There's a clue in the title
Student Solutions
Well done to the people who found the height (48) and then used it to get the triangle side lengths, but special congratulations to Michael from Taradale High School for a more direct approach using the powerful property that this arrangement contains three similar triangles.
If the yellow triangle has a hypotenuse, h, and if the yellow-orange triangle and the yellow triangle are similar, then the ratio, h : 100 is equal to the ratio 36 : h
From that we know that h must be 60, and using the same thinking again the hypotenuse of the orange triangle must be 80.
Teachers' Resources
This printable worksheet may be useful: Nicely Similar.
Why do this problem?
This problem gives students an insight into how powerful similarity can be in geometric problems as well as allowing students to practise identifying similar shapes and evaluating lengths in proportion. It leads students towards the reasoning that is central to trigonometry. It can also be approached using Pythagoras’ Theorem.
Possible approach
Start by asking students to observe the diagram.
How many right-angled triangles can they see?
If they call one angle in the diagram $\theta$ what can they say about other angles?
How many similar triangles can they see?
If necessary, recap or define basic properties of right-angled triangles and the conditions for two right-angled triangles to be similar.
Introduce the lengths (64, 36 and 100) and the question. Allow students to work in pairs or small groups. Encourage them to assign letters or symbols to unknown lengths and to find relationships between their unknowns, using what they know about similar triangles and/or Pythagoras’ Theorem. Encourage them to start working systematically with what they know. You could remind them that any rule they apply, they can apply to all three triangles.
As groups build up more equations, can they see any that they can solve or combine to find any of the missing lengths? Encourage them to share their ideas with each other and to experiment with sets of equations if they are not sure.
Once groups have found an answer, ask them to justify and consolidate their method. How many steps did they need? Which order did they use the information in? Can they explain their method in the simplest possible way?
At the end of the activity, bring the class together to share their methods. You might want to get the students to make posters explaining their methods as simply as possible.
Key questions
How many triangles can you see in the diagram? What have they got in common?
What does it mean for two triangles to be similar?
Can you write your method in fewer steps?
Is there another way of doing it?
Possible support
A warm-up exercise in finding missing lengths in similar triangles. Encourage students to compare a between-triangle scale factor method (the ratio of side A in triangle 1 to side A in triangle 2 is the same as the ratio between sides B in the two triangles) with a within triangle scale factor method (the ratio of side A to side B is the same in both triangles).
Alternatively, you could recap Pythagoras’ Theorem.
Possible extension
Do we have more information than we need?
If we have a right-angled triangle as shown in the diagram, how much information do we need to work out the value of the length of all sides in the diagram?
Can you make up your own question similar to this one?