P=12
Introduction
How many ‘different’ triangles can you make on a 3 by 3 geoboard show on the right? How do you know you have found all of them?
What properties does each triangle have? What is the ‘name’ of each triangle? What mathematical ways can you use to show that they have the same length? Same angle?
How many different sized angles are there in the different triangles? What have you noticed when you add up all the angles of each triangle? Click here for a hint.
What is the perimeter of each triangle? Which triangle gives you the biggest perimeter? Smallest?
What is the area of each triangle (if the area of a small square is 1 square unit)?
Constructing Triangles
The perimeter of a triangle is 12 units, and all of the side lengths are integers (whole numbers).
Come up with all the possible triangles that can be created using this rule.
How do you know which ones don't work?
How do you know if you have got all of them?
2, 5, 5 and 5, 2, 5 are congruent triangles. What do I mean by that?
Draw out all the different triangles, check that all your triangles work!
What you did just now was constructing SSS triangles. This is a rule that gives congruent triangles. What do you think SSS stands for?
Other ways of constructing congruent triangles are SAS, ASA, AAS and RHS. What do you think "A", "R" and "H" stand for?
Is SAS the same as SSA? Is ASA the same AAS? If not, how are they different? Will they give congruent triangles? Why/ why not?
SAS - Here is how to draw SAS triangles by hand or on mathspad and here is how to do it on geogebra.
ASA - Here is how to draw ASA triangles by hand or on mathspad and here is how to do it on geogebra.
For more interactive demonstrations and worksheets of interesting constructions, click here (mathopenref) or here (corbett maths videos 68-83).
Further Questions and Challenges
The 3, 4, 5 right-angled triangle is the simplest example of a Pythagorean Triple. This means it's a right-angled triangle where all three sides are integers. Can you find any more Pythagorean Triple?
What rule links the three sides of such triangles? Hint: you may want to square all three sides first and see if you can find a relationship between the the three numbers.
Try this constructing triangles task on nRich.
Back to the activity in the introduction, can you find all the quadrilaterals on the 3 by 3 geoboard?
Further Practice
Consolidate your understanding on constructing triangles by doing the relevant skills practice on DrFrostMaths, CorbettMaths, MyiMaths and Eedi. Watch any video and/or go through any online lesson as you see fit.
Transum
Constructions: Construct the diagrams from the given information then check your accuracy.