This article covers the important points and lines in a triangle, how they can be used to solve the common problems. Let us first learn the basic concepts of Perpendicular bisectors, angle bisector, median, incenter, circumcircle, orthocenter, centroid etc. The knowledge of these concepts are important to check on for solving some problems in qucik time frame.
Perpendicular Bisector & Circumcenter
Perpendicular Bisector : A straight line from center of a side of triangle such that it is perpendicular to the side.
Circumcenter : The point where the perpendicular bisector meet is called as circumcenter. If a circle is drawn with Radius eqaul to distance between circumcenter and vertex of the triangle, the other two vertex will be on the circumference of the circle. The circle thus formed is known as Circumcircle
Orthocenter & Altitudes
A straight line from vertex to the opposite side of the triangle such that it is perpendicular to that side of the triangle, this line is called as altitude.
Orthocenter: The point where all the three altitudes intersect is called as Orthocenter
Incenter: The point at which 3 angle bisector meets is known as incenter.
The largest circle that can be formed inside the triangle touching the sides is called incenter
Also for angle bisector divide the sides such that (for angle bisector Bb) BC/AB = Cb/cA
The radius of incircle is given as
Median of a Triangle and Centroid
Meidan: A striaght line through vertex to the opposite side such that it divides the side into 2 equal parts.
Centroid: Point where three meidan intersect is called as centroid
Important Points to Note from MBA Examination prespective:
1. If one draw a median it will divide the triangle into 2 equal triangles, area of each triangle will be half of original triangle
2. If all three medians are drawn it will divide the large triangle into 6 smaller triangles of equal areas
3. Centroid divided the median from vertex (say Point A) to the point of intersection (a) on the opposite side (BC) in the ratio of 2:1. i.e If centroid is m, Am: ma = 2:1
4. Centroid is also the centre of gravity of the the Triangle
Length of median
Most imp. (Theorem of Apollonius) = BC2 + BA2 = 2 x (median)2 + 2 x ( half of 3rd side)2
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