Question

ABC is a triangle and AD is median. If E is any point on AD, then

• A. $Ar\left(ABE\right)=Ar\left(ACE\right)$
• B. $BE=CE$
• C. $AB+BE=AC+CE$
• D. $AE=\frac{BE+CE}{2}$

Answer 1

The correct option is A $Ar\left(ABE\right)=Ar\left(ACE\right)$

ABC is the triangle and O the centroid. Let AO produced meet BC in D. The Area of the triangles ABD = Area of triangle ACD.
(Reason CD=BD and same altitude and Area = $\frac{1}{2}\times base\times altitude)$
Similarly, Area EBD = Area ECD. Subtracting i.e
Area ABD - Area EBD = Area ACD - Area ECD which implies Area ABE = Area ACE