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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Straight Lines

Olympiad workbook · IOQM · Straight Lines

Let A B C be a triangle in the x y plane, where B is at the origin...

Let

A B C

be a triangle in the

x y

plane, where

B

is at the origin

(0,0)

. Let

B C

be produced to

D

such that

B C: C D=1: 1, C A

be produced to

E

such that

C A: A E=1: 2

and

A B

be produced to

F

such that

A B: B F=1: 3

. Let

G(32,24)

be the centroid of the triangle

A B C

and

K

be the centroid of the triangle

D E F

. Find the length

G K

.

Let A B C be a triangle in the x y plane, where B is at the origin (0,0) . Let B C be produced to D such that B C: C

Correct answer

40

Step-by-step solution

Let A B C be a triangle in the x y plane, where B is at the origin (0,0) . Let B C be produced to D such that B C: C

\because B C: C D=1: 1 \quad

hence

D=\left(2 x_1, 2 y_1\right)

C A: A E=1: 2 \quad

hence

E=\left(3 x_2-2 x_1, 3 y_2-2 y_1\right)

A B: B F=1: 3 \quad

hence

F=\left(-3 x_2,-3 y_2\right)

\therefore \quad

Centroid of

\triangle D E F=K=(0,0)

Centroid of

\triangle A B C=G=(32,24) \therefore \quad G K=\sqrt{32^2+24^2}=40

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