# Triangle midpoints

*Triangle Midpoints printable sheet*

**Given the three midpoints of the sides of a triangle, can you find a way to construct the original triangle?**

For example:

Choose any three points.

Can you construct a triangle such that your three points are the midpoints of its sides?

Is there more than one possible triangle for any given set of midpoints?

Draw any triangle, find the midpoints, and join them.

What do you notice?

How can this help you solve the problem in reverse, starting with the midpoints and constructing the triangle?

We received two different approaches to the solution. The first is based on what Stephanie from Beecroft School noticed.

Stephanie noticed that the large triangle is made from four congruent triangles, one of which is the triangle joining the midpoints. Stephanie therefore created the larger triangle by making three copies of the smaller one and translated and rotated them into place.

My question is:

How do you know the triangles are congruent? Does the following diagram help?

Chip at King's Ely School and Eli sent in a solution based on Stephanie's ideas.

Like my question above, I want to know how you know the vertices of the larger triangle can be found in this way? Also, if you can convince me you are right, how does the construction to find the midpoint of the sides of the triangle work and why?

The second approach involved parallel lines:

The original triangle PQR can be drawn by constructing a line through A parallel to BC, a line through B parallel to AC and a line through C parallel to AB.

You should then be able to prove that the points A, B and C are the mid-points of PR, PQ and QR.

How do you know the sides of the larger triangle are parallel and how can you construct each of the lines parallel to the sides of the original triangle. Why does the construction work?

Can you tell us more about how these ideas work please?

### Why do this problem ?

### Possible approach:

*This printable worksheet may be useful: Triangle Midpoints.*

To start with there is some thinking to do about the way in which knowledge of the mid points is sufficient to fix the triangle (surprising?). Show the question and ask for ideas about possible approaches, things the group notices and things that might be true but they/others would need to be convinced about.

- What does it look like?
- Is it regular?

- What do you notice? The answer might just be they they make a triangle but press for any other ideas which might emerge that you could pursue.

Use time to discuss what different groups notice and how it might help. Encourage rigour.

This problem creates a good opportunity to practise some common classroom geometry constructions, for example: a perpendicular bisector of a given line, angle bisector centre for the circumcircle of a given triangle, drawing a line through a given point parallel to a given line.

It is particularly important to promote a lively discussion of possible reasons for a construction method's validity.

### Key questions:

- What does working backwards from what you know reveal about the problem that could be helpful?
- How do you know...? How can you convince us?
- Why does the construction work ?

### Possible support:

Establish the relationships with paper folding and cutting and use this to see why triangles are congruent and therefore why particular approaches are valid.