1. In triangle ABC , O is the point of intersection of the medians AD, BE and CF such that area of triangle AOB, AOC and BOC are equal. If OA2+OB2+OC2=k(a2+b2+c2) ; a, b , c being the sides opposite to BC,CA and AB respectively, then which of the following is true? A.k<1/3 B.k=1/3 C.1/3 <k<1/4 D. k= 1/4 E. k>1/4 Plz explain the theorems, whichever u apply |
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s shahid basha
In a triangle ABC if AD, BE and CF are medians then
4(AD^2 + BE^2 + CF^2) = 3(a^2 + b^2 + c^2)
now the intersection of median is the centroid which divides the median in the ratio of 2:1, 2 being
from the vertex. So OA = 2/3AD, OB = 2/3BE, and OC = 2/3CF.
Substituting in the above equ we get
OA^2 + OB^2 + OC^2 = 1/3 ( a^2 + b^2 + c^2).
therefore k = 1/3.
If i'm wrong pls correct me
thank u
shahid.
n/a
hello
yes ,ur answer is correct.
thanks for ur help, but i could not get the relationship 4(AD^2 + BE^2 + CF^2) = 3(a^2 + b^2 + c^2) .
Moreover the statement from the question a, b , c being the sides opposite to BC,CA and AB respectively are also not clear.can u please explain.
thanks a lot.
true,friend!d info in d question isnt clear.how can one spot an opposite side to one side in a triangle.hence,d info dat a,b,c is not clear.
s shahid basha
@aneet
a,b, and c are opposite sides of BC, AC and AB.
These are standard notations in geomentry. Here A, B, and C are vertices of a triangles. The line joining the
two vertices A and B is AB. Now the according to standard notation the side opposite the side AB is generally
written as 'c'. Similarly the side opposite to line BC is 'a' and the line opposite to line CA is 'b'.
Now coming to the formula.
If a tri ABC has medians on to their opp sides then three times there sum of square of sides is equal
to four times the sum of their medians. It's also a standard formula.
Hope i cleared ur doubts. If u still didn't understand feel free to ask me again.
thank u
shahid.
n/a
hello
thanks sir 4 clearing my doubts. does the relationship 4(AD^2 + BE^2 + CF^2) = 3(a^2 + b^2 + c^2) holds for the medians individually also i.e 4(AD^2)=3(a^2).
@aneet
No, the relationship does not holds good individually.
There is another relationship between sides and median of a triangle.
If ABC is a triangle and 'D' is the midpoint of side BC which is opposite to the vertex A, then AD would be the median. Then
AC^2 + AB^2 = 2( AD^2 + CD^2 ).
hope that this would help u.
thank u
shahid.
n/a
hello
thanks a lot sir.
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