Geometry - Parallelogram Properties

Question

GRAM is a parallelogram. RM intersects GA at point S. The perimeter of triangle GRS is 21. The perimeter of triangle RAS is 26. The perimeter of triangle GRA is 35. What is the length of segment RM? Please explain how to do step by step. Thanks

Answer 1

The diagonals of a parallelogram bisect each other.

GR+RS+SG=21

RA+AS+RS=26

AS=SG because the diagonals of a parallelogram bisect each other.

substitute

GR+RS+AS=21

RA+RS+AS=26

add the two equations

GR+RA+2RS+2AS=47

now for the third triangle...

GR+RA+AG=35 but AG=2AS because diagonals of a parallelogram bisect each other

so we have GR+RA+2AS=35

GR+RA+2RS+2AS=47

GR+RA        +2AS=35

subtract the equations

  0+0   +2RS+0   =12

2RS=12 (we can basically stop here because we have the answer, however...)

RS=12/2

RS=6

RS=SM (diagonals of a...)

RM=RS+SM

RM=6+6

RM=12