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How can i prove that the medians of a triangle are concurrent for a class 9 standart?


Asked by ishitahazarika.i22 13th January 2019, 12:45 AM
Answered by Expert

Let D and E are mid point of side AC and side AB respectively. Let us join D and E as shown in left side of figure.
We know that since D and E are mid points two sides of triangle, line DE is parallel to third side BC and DE = (1/2)BC.
Since DE is parallel to BC, as marked in the figure, begin mathsize 12px style angle end styleEDG = begin mathsize 12px style angle end styleGBC and begin mathsize 12px style angle end styleDEG = begin mathsize 12px style angle end styleGCB .
Hence ΔGED and ΔGBC are similar triangles. we have begin mathsize 12px style fraction numerator E D over denominator B C end fraction space equals space fraction numerator E G over denominator G C end fraction space equals fraction numerator D G over denominator G B end fraction equals 1 half end style
Hence the intersection point G divides the median BD and CE in the ratio 2:1  or GD = (1/3)BD  and GE = (1/3)CE
Now if we draw medians BD and AF as shown in right side of figure, we can similarly prove ΔGFD and ΔGAB are similar,
BD and AF intersects at G so that GF = (1/3)AF  and GD = (1/3)BD
since we have proved in both the cases,  if any two medians intersect, then intersection point divides the medians in the ratio 1:3,
hence all the three medians are concurrent
Answered by Expert 14th January 2019, 1:59 PM
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