New Math Drill (Level 2) : Geometry (Plane figure) Problem3(2) NAKANO ATTACHED TO MEIJI UNV.-2014

As shown in the figure below, there is a trapezoid ABCD, and AD and BC are parallel, and AD = 2cm, BC = 3cm. 

P and Q are points on the side AB, DC and AP : PB = DQ : QC = 2 : 1. 

R is an intersection of PQ and AC. 　

S is an intersection of CD and the extended line of BR. 

T is an intersection of the extended lines of BR and AD. 

In this case, find DS : SQ : QC by the ratio of the simplest integer.

Answer

12 : 2 : 7

Solution

AR : RC = AP : PB = 2 : 1 

△ART and △CRB are homothetic amd homothetic ratio is 2 : 1. 

AT = 3cm × 2 = 6cm 

OT = 6 - 2 = 4cm 

DS : SC = DT : CB = 4 : 3 

DC is set to to be (4 + 3) × (2 + 1) = 21, DS : SQ : QC = 12 : 2 : 7 as shown below.