Triangles – Four Centres

The triangle was one of the most investigated shapes. Hence many concepts related to triangles have been discovered. We will touch upon a few of them.

All human cultures discovered very early that the triangle is the most rigid structure. It is also the simplest structure that needs only 3 sticks and some rope!

Area & Perimeter

Two of the most important and basic properties of a triangle (and indeed any closed figure are perimeter and area. The perimeter of a triangle is the sum of the lengths of the 3 sides.

As we will see in the chapters (25.2 & 25.3) on measurement of area, the area of a triangle can be worked out by finding a rectangle of the same area. We can discover such an area also by folding a paper triangle. (Chapter 25.3)

The required rectangle turns to have one side as the base of the triangle and the other side as half the height of the triangle. (This can also be seen as a rectangle with one side being half the base and the other side equal to the height.). Hence the formula of the area of any triangle can be written as BH where B is the base of the triangle and H is the height of the triangle. Height of a triangle is the measure of the Altitude which can be drawn from the vertex opposite the base to the base. This formula requires two different lengths to be measured.

There is another formula which connects the area to the semi-perimeter (half the perimeter) of a triangle to its area. This was discovered in the year 60 CE by Hero of Alexandria and requires only one length! We will formally study this formula only in middle school!

Centres of a Triangle

The following 4 sets of lines can be drawn for any triangle. Each set contains 3 lines because the triangle has 3 sets of sides and angles. These are perpendicular bisector, angular bisector, altitude and median. It is also an interesting property of a triangle that each of these sets of 3 lines meet at a point which is called by the name with a suffix - centre.

1. Perpendicular Bisectors of each side are lines which bisect the lines and which are perpendicular to them. They meet at the Circumcentre

2. Angular Bisectors of each angle bisect the angles. They meet at the Incentre

3. Altitudes are lines which are perpendicular to a side and which also pass through the opposite vertex. They meet at the Orthocentre

4. Medians are lines which join each vertex with the mid-point of the opposite side. They meet at the Centroid.

Each of these points themselves have interesting properties which we will not go into now. These special lines & points can easily be studied informally by just using broom sticks!