1. G is the centroid of equilateral Triangle ABC. D,E, and F are midpointsof the sides as shown. P,Q, and R are the midpoints of line AG,line BG and line CG, respectively. If AB= sqrt 3, what is the perimeter of DREPFQ?

xxJenny1213xx Nov 17, 2019

#2**+2 **

Since ABC is isosceles....Angle FAP = Angle EAP = 30°

EA = FA

PA = PA

So....by SAS....triangle FAP = triangle EAP

And Angle EAP = Angle ECR = 30°

So angle CGA = 120°

Draw DF and angle DQF = 120°

And angle QDF = angle QFD = 30°

So triangles CGA and DQF are similar

Draw GE

And EA = sqrt (3) / 2

And triangle GEA is a 30 -60 -90 right triangle

So.....AG = EA * 2 * sqrt (3) = sqrt (3) / 2 * 2 / sqrt (3) = 1

And triangle BDF is similar to triangle BCA

And since BF = (1/2) BA, then DF = (1/2)CA

And since CGA and DQF are similar....then DF = (1/2) CA so QF = (1/2) AG = 1/2

And AG is the side of regular hexagon DREPFQ

So.....the perimeter is 6 (AG) = 6 (1/2) = 3

CPhill Nov 17, 2019

#3**+1 **

YES! That is what I was trying, proving that there are 30 60 90 triangles within the shape, then solving!

CalculatorUser Nov 17, 2019