Question

ABC is a triangle in which P is the midpoint of AB and Q is a point on AC such that AQ = 2QC .A line through P parallel to BQ meets AC in R prove that AR = QC​

Answer 1

Answer:

Step-by-step explanation:

Given A △ABC in which P is the mid-point of BC, Q is the mid-point of BC, Q is the mid-point of AP, such that BQ produced meets AC at R

To prove RA=

3

1

​

CA

Construction Draw PS||BR, meeting AC at S.

Proof In △BCR, P is the mid-point of BC and PS||BR.

∴ S is the mid−point of CR.

⇒ CS=SR

In △APS, Q is the mid-point of AP and QR||PS.

∴ R is the mid−point of AS

⇒ AR=RS

From (i) and (ii), we get

AR=RS=SC

⇒ AC=AR+RS+SC=3AR

⇒ AR=

3

1

​

AC=

3

1

​

CA [Hence proved]