For which value of x is parallelogram upper X upper Y upper Z upper A a rectangle?
Quadrilateral upper X upper Y upper Z upper A is shown with diagonals upper X upper Z and upper A upper Y intersecting in the middle of the figure. The part of the diagonal between upper A and the intersection of the diagonals is labeled 5 x minus 5. The part of the diagonal between upper Z and the intersection is labeled 3 x plus 3.
(1 point)
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To determine when the parallelogram is a rectangle, we need to find the value of x such that the diagonals are perpendicular to each other.
Let's use the fact that the diagonals of a parallelogram bisect each other. Since the part of the diagonal between A and the intersection is labeled 5x - 5, the other part has the same length. Therefore, the entire diagonal AX is 2(5x - 5) = 10x - 10.
Similarly, the part of the diagonal between Z and the intersection is labeled 3x + 3, so the other part is also 3x + 3. Therefore, the entire diagonal ZY is 2(3x + 3) = 6x + 6.
For the diagonals to be perpendicular, their slopes must be negative reciprocals of each other. The slope of diagonal AX is (change in y / change in x) = (-10) / 10 = -1, and the slope of diagonal ZY is (change in y / change in x) = (-6) / 6 = -1.
Since the slopes are equal and negative reciprocals, the diagonals are perpendicular when the slopes are -1.
To find the value of x, we can set their slopes equal to -1:
(-10) / 10 = (-6) / 6
Simplifying, we get:
-1 = -1
Since this equation is always true, the value of x can be any real number. Therefore, there is no specific value of x for which the parallelogram is a rectangle.