Let's analyze the given quadrilateral WXYZ to find the length ZX. We are told that the figure is a rectangle.
First, by the Segment Addition Postulate, we can express the length of each diagonal as the sum of lengths of its smaller segments.
ZX= ZP+PX and WY=WP+PY
Additionally, because a rectangle is a parallelogram, we know that the diagonals bisect each other. This means that point P divides each diagonal into halves.
ZP= PX and WP= PY
We can use this fact to rewrite the Segment Addition Postulate equations in terms of the given expressions, ZP= 4x-9 and PY= 2x+5.
Equation
ZX=ZP+PX
WY=WP+PY
Substitution
ZX=ZP+ ZP
WY= PY+PY
Simplification
ZX=2ZP
WY=2PY
Substitution
ZX=2( 4x-9)
WY=2( 2x+5)
Now, recall that the diagonals of a rectangle are congruent. Therefore, their lengths are equal.
ZX=WY ⇔ 2(4x-9)=2(2x+5)
We can use this equation to solve for x.
