+614 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?
+129899 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
+2864 YES! That is what I was trying, proving that there are 30 60 90 triangles within the shape, then solving!
