Question

In ∆ABC ,point D is midpoint of side BC . Point E is midpoint of median AD. Ray BE intersect side AC at point P .
Prove that: AP=1÷3×AC

Answer 1

AP = 1/3 AC. (Proved)

Step-by-step explanation:

See the attached diagram.

We extend the side AB to R such that AB = AR.

Now, considering Δ BRC, D is the midpoint of DC and A is the midpoint of BR, so AD is parallel to CR.

So, as E is the midpoint of AD then Q will be the midpoint of CR.

So, BQ is a median of Δ BRC and again, as A is the midpoint of BR, so CA will be another median of Δ BRC.

Now, BQ and CA meet at P and hence P is the centroid of Δ BRC.

So, AP = 1/3 AC. (Proved)

{Since centroid divides the median in 1 : 2 ratio}