Starter of the Day

Bizarre Triangle

Bizarre Triangle

If the base of this triangle became 8cm but the other sides remained 5cm, how much would its area increase?



Topics: Starter | Area | Mensuration | Problem Solving | Pythagoras

  • Transum,
  • At the time of writing this is a new starter and not many people have had a chance to score it. Though it is going to be perhaps out of reach of the younger pupils, those with a knowledge of Pythagoras' Theorem should find it quite easy. The height of the triangle works out to be a whole number in both the before and after situations and from that, the surprising result can be found. A complete explanation can be found in the answers below. Please feedback any thoughts on this.
  • Dave, Australia
  • It is unclear whether the other lengths of the triangle adjust along with the base or if they remain constant. If they (the 5cm sides) remain constant the area will be unchanged - however if they do not the area will be increasing...
  • Joe Young, Chirstchurch
  • This doesn't help, it doesn't really tell us what to do. Also it gives no place to answer and find out if we have even got it right. My students are unsure what to do.

    [Transum: Thanks for your feedback Joe. This activity is designed as a lesson Starter to be projected in front of a class at the beginning of a lesson. The teacher can then lead a discussion with pupils about what the solution might be. The answer appears lower down this page for teachers who are signed in.]
  • Elliot Oliver, Temuka
  • Rephrasing what Dave and Joe said, this starter isn't the best. It is not very clear about if the hypotenuses of the two triangles stay the same, as well as if the height changes. To clarify that would be great.
  • Yolax, Australia
  • I must disagree with the critical comments this is a beaut starter it clearly states that the 5cm sides remain the same. The area will not change, so that is the learning moment revel in it!

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Can you find another pair of triangles that, like those above, surprisingly have the same areas?

What is the length of the base which gives the triangle mentioned above the maximum area?

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Laptops In Lessons

Teacher, do your students have access to computers such as tablets, iPads or Laptops?  This page was really designed for projection on a whiteboard but if you really want the students to have access to it here is a concise URL for a version of this page without the comments:

However it would be better to assign one of the student interactive activities below.

Laptops In Lessons

Here is the URL which will take them to a related student activity.

Student Activity


Extension Answers

A first thought might be that the graph of area against length of base is symetrical but it is not. A graphic calculator will help you identify equal-area bases and the length of base giving the maximum area:

Bizarre Triangle on a calculator


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