Let's analyze the given quadrilateral to find the variables x and y. Keep in mind, we have been told that the figure is a rectangle.
By the Segment Addition Postulate, we can express each diagonal as the sum of its smaller segments.
FH= FM+MH and GJ=GM+MJAdditionally, because a rectangle is a parallelogram, we know that the diagonals bisect each other.
FM= MH and GM= MJ
We can use this fact to rewrite the Segment Addition Postulate equations in terms of the given expressions, FM= 3x+y and GM= 13.
Equation
FH=FM+MH
GJ=GM+MJ
Substitution
FH=FM+ FM
GJ=GM+ GM
Simplification
FH=2FM
GJ=2GM
Substitution
FH=2( 3x+y)
GJ=2( 13)
Recall that the diagonals of a rectangle are congruent. Therefore, their lengths are equal.
FH=GJ ⇔ 2(3x+y)=2(13)
Because our quadrilateral is a rectangle, both opposite sides are congruent. We can use the given expressions to write one more equation.
GH=FJ ⇔ 11=- 3x + 5y
Let's create a system of equations using both equations above.
2(3x+y)=2(13) 11=-3x+5y
To solve it we will use the Substitution Method.
