Question

In the given figure, G is the mid-point of the side PQ of  ΔPQR and GH || QR. Prove that  H is the mid point of the side PR of the triangle PQR.

Answer 1

Given, G is the mid-point of the side PQ of \(\triangle \)PQR,

\(\therefore\) PG = GQ or PQ = 2PG   .....(1)

Now, since GH || QR.

Using Basic Proportionality theorem, we get,

\(\frac{PG}{GQ} = \frac{PH}{HR}\)

\(\frac{PG}{PQ} = \frac{PH}{PR}\)

⇒ \(\frac{PG}{2PG} = \frac{PH}{PR}\)    [Using (1)]

⇒ \(\frac12= \frac{PH}{PR}\)

⇒ PR = 2PH

⇒ PH + HR = 2PH

⇒ HR = PH

Hence, H is the mid-point of PR.

Answer 2

Since G is the mid-point of PQ,

:.  PG = GQ 

or,    PG / GQ = 1

According to the question,

GH || QR