In this exercise will we work with triangles. [br]Consider the following triangles given by the length of its sides:[br][br]Triangle1: sides are 5 - 5 - 8[br]Triangle2: sides are 5 - 5 - 6[br][br]Which one you think is bigger? [br]Here we are thinking about the area of each triangle.[br]Before we continue, try to make a guess... What do you think?

We have the following triangles given by the length of its sides:[br][br]Triangle1: sides are 5 - 5 - 8[br]Triangle2: sides are 5 - 5 - 6[br][br]We were asked to think which one of this has bigger area.[br]How did it go? Could you solve it?[br]What reasoning allowed you to solve it? how did you realize?[br][br]Here we present a possible solution that visually is very clear.

Start by drawing the height of each triangle. We can use Pythagoras theorem to find out the height size in this isosceles triangle, since we get right-angled triangles. For example, lets take a look to Triangle1:[br]

With the measure of this segments we can calculate the areas of the triangles:[br][math]Area=\frac{1}{2}\times base\times height[/math][br]For Triangle1[br][math]Area1=\frac{1}{2}\times8\times3=12[/math][br]For Triangle2[br][math]Area2=\frac{1}{2}\times6\times4=12[/math][br]So we can check that they have the same area.

There is a very nice and geometric way to realize this whitout doing any calculation. In fact, we can think of each of this triangles as two copies of a smaller triangle with sides 3 - 4 - 5:

As we can appreciate, when t = 90° we get a copy of Trianlge 2, while when t = 180° we get a copy of a simetric Triangle 1. [br]The fact that rotation of the triangles does not modify the area, we conclude that both triangles has the same surface.