P, Q, and R are on A B, B C, and A C of the equilateral triangle A B C, respectively. A P: P B=C Q: Q B=1: 2. G is the centroid of the triangle P Q B and R is the mid-point of A C. Find B G: G R.

Question

P, Q, and R are on A B, B C, and A C of the equilateral triangle A B C, respectively. A P: P B=C Q: Q B=1: 2. G is the centroid of the triangle P Q B and R is the mid-point of A C. Find B G: G R. (A) 1: 2 (B) 2: 3 (C) 3: 4 (D) 4: 5

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/1e4acf5f87f37a39991767190858641a-.jpeg" /> Let A B=B C=A C=3 x ∴ B P=B Q=P Q=2 x (∵ A P: B P=C Q: B Q=1: 2) As triangle B P Q and triangle B A C are equilateral triangle, the centroid of triangle B P Q lies on B R (where B R is median drawn on to A C ) We know that centroid divides the median in the ratio 2: 1. ∴ B G=(2/3) ([√3(2 x)]/2)=(2 √3 x/3) But B R=(√3(3 x)/2)=(3 √3 x/2) Now G R=B R-B G =(3 √3 x/2)-(2 √3 x/3)=(5 √3 x/6) Now B G: G R=(2 √3 x/3): (5 √3 x/6)=4: 5.

Q. $P, Q$ , and $R$ are on $A B, BC$ , and $A C$ of the equilateral triangle $A BC$ , respectively. $A P : PB = CQ : QB = 1$ : 2. $G$ is the centroid of the triangle $PQB$ and $R$ is the mid-point of $A C$ . Find $BG : GR$ .

$1 : 2$

$2 : 3$

$3 : 4$

$4 : 5$

Q. P,Q, and R are on AB,BC, and AC of the equilateral triangle ABC, respectively. AP:PB=CQ:QB=1 : 2. G is the centroid of the triangle PQB and R is the mid-point of AC. Find BG:GR.

1:2

2:3

3:4

4:5

Q. $P, Q$ , and $R$ are on $A B, BC$ , and $A C$ of the equilateral triangle $A BC$ , respectively. $A P : PB = CQ : QB = 1$ : 2. $G$ is the centroid of the triangle $PQB$ and $R$ is the mid-point of $A C$ . Find $BG : GR$ .

$1 : 2$

$2 : 3$

$3 : 4$

$4 : 5$