Question

isosceles triangles BCX,CAY, and ABZ are constructed externally on the sides of a triangle ABC. show that the centroids of triangle ABC and triangle XYZ coincide

Answer 1

Using vector geometry we can solve this question easily.by using the facts

1.the midpoint of a segment is average of that two vectots

2.the cenroid of a triangle is the average of that yhree vectors

solution . Centrad 13 a point where all the three medians of a torangle intersect. so the centooid of a torangle can be found by the average of x-co-ordirale's value and y-coordinate's value. Centraid of tarangle ABC of points A,B,C 15 (x1+ x2 + x& )/3, ( Ys+Y2 + Y3)/3 It is given that the isoceles za torangles Bex, CAÝ, ABZ are Constructed extrenally on the sides oj A ABC. .. we have to show that the ... centroids of AABC + Axyz co-incide. Y . Let or be the centraid of A ABC and on be the cenbord CScannedbwith CamScanner co OLLA xyz. to show 0,5 02

See Using vectors, From vector geometry; 1) The midpoint of the segment AB is the average of - Vectors à and B. 2) The centaid of triangle A ABC is the average of three vectoss AB and a C - Here, Å is the average of 2 and y uely, B is the average of 2 and 3: Z is the average of 7 and X Oi, since it is a median, is the average of Ar Band 2 and I, I and e that means It is the average of ř and 7 and ? That means the centroid of oxy.z7s equal to centroid of AABC ie, Oi=02 Hence proved, ned with CamScanner