Close to triangular
Drawing a triangle is not always as easy as you might think!
Age
14 to 16
Challenge level
Problem
Here are the coordinates of nine points. It is possible to draw a triangle so that the shortest distance from each point to the triangle is at most one unit.
$(0, 0)$
$(8, 2)$
$(7, 8)$
$(170, 180)$
$(340, 360)$
$(2001, 1000)$
$(1500, 750)$
$(3000, 2000)$
$(4002, 2000)$
Can you find a suitable triangle? Is there more than one possibility?
Given three points, it is always possible to draw different triangles with edges passing through those three points - here are some examples of triangles going through the same three points:
Image
Here are some examples of different triangles going through the same set of four points:
Image
Is it always possible to draw triangles through a set of four points, whatever their position?
Investigate some examples and explain your findings.
What happens when we try to draw triangles through five points?
Getting Started
Drawing a sketch really helps!
Student Solutions
Preveina from Crest Girls' Academy made a start on this problem:
For three points, there are always infinitely many such triangles because every time you extend the length of the lines in a triangle you will be making a new point; by doing this you'll be producing unique triangles every time. This then leads on having infinity triangles made.
The picture below shows a sequence of triangles - the black lines pass through two of the points, and a variety of lines can pass through the third point, extending one of the lines in the original triangle.
Image
Preveina went on to show some examples of configurations of four and five points where a triangle could be drawn.
To consider whether all configurations are possible, consider the set of points below:
Image
Can you find a way to draw a triangle passing through all four points? Can you convince yourself it is impossible?
Teachers' Resources
Why do this problem?
This problem is in two parts, which can be attempted independently of each other. The first part requires students to apply their knowledge of coordinate geometry and the equations of straight lines. The second part explores triangles passing through four points and encourages conjecture about the necessary conditions for this to be done.
The problem lends itself to collaborative working, both for students who are inexperienced at working in a group and students who are used to working in this way.
Many NRICH tasks have been designed with group work in mind. Here we have gathered together a collection of short articles that outline the merits of collaborative work, together with examples of teachers' classroom practice.
Possible approach
This is an ideal problem for students to tackle in groups of four. Allocating these clear roles (Word, pdf) can help the group to work in a purposeful way - success on this task should be measured by how effectively the members of the group work together as well as by the solutions they reach.
Introduce the four group roles to the class. It may be appropriate, if this is the first time the class have worked in this way, to allocate particular roles to particular students. If the class work in roles over a series of lessons, it is desirable to make sure everyone experiences each role over time.
For suggestions of team-building maths tasks for use with classes unfamiliar with group work, take a look at this article and the accompanying resources.
Hand out either or both of the task sheets (First task: Word, pdf. Second task: Word, pdf.) to each group, and make it clear that everyone needs to be ready to
share what they did with the rest of the class at the end of the sessions. Exploring the full potential of this task is likely to take more than one lesson, with time in each lesson for students to feed back ideas and share their thoughts and questions.
You may want to make squared or graph paper, poster paper, and coloured pens available for the Resource Manager in each group to collect.
While groups are working, label each table with a number or letter on a post-it note, and divide the board up with the groups as headings. Listen in on what groups are saying, and use the board to jot down comments and feedback to the students about the way they are working together. This is a good way of highlighting the mathematical behaviours you want to promote, particularly with a
challenging task such as this.
You may choose to focus on the way the students are co-operating:
Group A - Good to see you sharing different ways of thinking about the problem.
Group B - I like the way you are keeping a record of people's ideas and results.
Group C - Resource manager - is there anything your team needs?
Group B - I like the way you are keeping a record of people's ideas and results.
Group C - Resource manager - is there anything your team needs?
Alternatively, your focus for feedback might be mathematical:
Group A - I like the way you chose to represent the situation with a diagram.
Group B - You've drawn some straight lines - would it help to express them algebraically?
Group C - Good to see that someone's checking that each point is close enough to the triangle.
Group B - You've drawn some straight lines - would it help to express them algebraically?
Group C - Good to see that someone's checking that each point is close enough to the triangle.
Make sure that while groups are working they are reminded of the need to be ready to present their findings at the end, and that all are aware of how long they have left.
We assume that each group will record their diagrams, reasoning and generalisations on a large flipchart sheet in preparation for reporting back. There are many ways that groups can report back. Here are just a few suggestions:
- Every group is given a couple of minutes to report back to the whole class. Students can seek clarification and ask questions. After each presentation, students are invited to offer positive feedback. Finally, students can suggest how the group could have improved their work on the task.
- Everyone's posters are put on display at the front of the room, but only a couple of groups are selected to report back to the whole class. Feedback and suggestions can be given in the same way as above. Additionally, students from the groups which don't present can be invited to share at the end anything they did differently.
- Two people from each group move to join an adjacent group. The two "hosts" explain their findings to the two "visitors". The "visitors" act as critical friends, requiring clear mathematical explanations and justifications. The "visitors" then comment on anything they did differently in their own group.
Key questions
If your focus is effective group work, this list of skills may be helpful (Word, PDF). Ask learners to identify which skills they demonstrated, and which skills they need to develop further.
If your focus is mathematical, these prompts might be useful:
Does it help to draw a diagram?
What happens to the equation of a line when you move it slightly?
How does moving the lines affect the distances from each point to the lines?
For Part 2: Can you draw a configuration of four points which can NEVER lie on the edges of a triangle?
Is everyone in your group convinced that it is NEVER possible?
What is special about the position of the points?
Possible extension
Providing reasoned arguments for when it is and isn't possible to draw a triangle through four, five or even six points is a very challenging mathematical activity!
The stage 5 problem Erratic Quadratic uses the idea of finding an object which passes close to a set of points, but uses curves instead of straight lines.
Possible support
By working in groups with clearly assigned roles we are encouraging students to take responsibility for ensuring that everyone understands before the group moves on.
The game Diamond Collector may offer useful practice at working out the equations of straight lines through points in advance of tackling the first part of this problem.