Class 10: Triangles – Lecture Notes – ICSE / ISC / CBSE Mathematics Portal for K12 Students

Centroid

The point of intersection of the three medians is called the centroid of the triangle.

In the adjoining figure, $G$ is the centroid.

$AP, BQ \ and \ CR$ are the medians which divide the corresponding sides $BC, AC \ and \ AB$ respectively in two equal halves.

Hence, $BP=PC, CQ=QA \ and \ AR=RB$

The centroid of the triangle always divide each of the medians in the ratio of $2:1$

Therefore, $AG:GP = 2:1,$ $BG:GQ=2:1 \ and \ CG:GR=2:1$

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Incentre

The point of intersection of the bisectors of the internal angles of a triangle is called the Incentre of the triangle.

The Incentre of the triangle is equidistant from each of the sides of the triangle.

Hence $IF=IE=ID$

If you draw a circle, with $I$ as the center, then the radius of this Incircle would be $IF \ or \ IE \ or \ ID$

In the adjoining figure, $AI, BI \ and \ CI$ are bisectors of angles $A, B \ and \ C$ respectively.

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Circumcenter

The point of intersection of the perpendicular bisectors of the three sides is the circumcenter of the triangle. In the adjoining diagram, you can see that $BD=CD, CE=AE \ and \ AF=CF$ .

The distance from the center to the three vertices $A, B \ and \ C$ are equal. i.e. $AO = BO=CO$

If you draw a circle with $O$ as the center, and the radius $AO \ or \ BO \ or \ CO$ , the circle will encircle the triangle and touch the three vertices.

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Orthocenter k41

The point where the three perpendiculars drawn from the vertices of a triangle to the opposite side of the triangles meet is called the orthocenter of the triangle.

In the adjoining figure, $AD \perp BC, AD \perp BE \ and \ CF \perp AB$