Question

In triangle ABC. P is the mid-point of side BC. A line through P and parallel to CA meets AB at point Q: and a line through Q and parallel to BC meets median AP at point R. Hence, $BC=4QR$
State whether the above statement is true or false.

• A. True
• B. False

Answer 1

The correct option is A True

Above statement is true.

Reason:

Given: In $△ABC$, $P$ is the mid-point of $BC$, $PQ\parallel CA$, $PQ$ meets $AB$ in $Q$, $QR\parallel BC$, $QR$ meets $AP$ in $R$.

To prove: $BC=4QR$

Proof: In $△ABC,P$ is the mid-point of $BC$ and $PQ\parallel AB$.

$\therefore Q$ is the mid-point of $AB$[Converse of mid-point theorem]

In $△ABP,Q$ is the mid-point of $AB$ and $QR\parallel BP$.

$\therefore R$ is the mid-point fo $AP$[Converse of mid-point theorem]

$\Rightarrow AP=2AR$

In $△ABP,Q$ is the mid-point of $AB$ and $R$ is the mid-point of $AP$.

$\therefore QR=\frac{1}{2}BP$[Mid-point theorem]

$\Rightarrow QR=\frac{1}{2}\left(\frac{1}{2}BC\right)$[$P$ is the mid-point of $BC$]

$\Rightarrow QR=\frac{1}{4}BC$

$\Rightarrow BC=4QR$

So, $\text{A}$ is correct option.