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Answer :
Final answer:
The slope of WZ and the slope of a side adjacent to WZ is -9/4.
The length of WZ is √97 and the length of a side adjacent to WZ is √41.
D. By calculating the slopes and lengths, parallelogram WXYZ is determined not to be a rectangle, rhombus, or square because it does not have perpendicular consecutive sides or equal side lengths.
Explanation:
To determine if parallelogram WXYZ is a rectangle, rhombus, square, or none of these, we need to find the slopes and lengths of its sides.
(a) Find the slope of WZ and the slope of a side adjacent to WZ.
To find the slope of WZ, we use the slope formula (y2 - y1)/(x2 - x1). For W(-1, 2) and Z(3, -7), the slope is (-7 - 2)/(3 - (-1)) = -9/4.
For the slope of side WX, adjacent to WZ, with W(-1, 2) and X(-5, 7), the slope is (7 - 2)/(-5 - (-1)) = 5/-4 = -5/4.
(b) Find the length of WZ and the length of a side adjacent to WZ.
The length of side WZ is found using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2). The length of WZ is √((3 - (-1))^2 + (-7 - 2)^2) = √(4^2 + (-9)^2) = √(16 + 81) = √97.
For the length of side WX, it is √((-5 - (-1))^2 + (7 - 2)^2) = √(-4^2 + 5^2) = √(16 + 25) = √41.
(c) What can we conclude about parallelogram WXYZ?
A rectangle has consecutive sides perpendicular (slopes are negative reciprocals), and a rhombus has all sides equal length. Since no sides are perpendicular (slopes -9/4 and -5/4 are not negative reciprocals), WXYZ is not a rectangle. Also, the lengths are not equal (√97 is not equal to √41), so it's not a rhombus or square. Therefore, D. WXYZ is none of these.
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Rewritten by : Barada
a. Slope of WZ = -2.25; Slope of WX = 5
b. WZ = √97; WX = √41
c. WXYZ is not a rectangle, rhombus, nor a square. We can conclude that: D. WXYZ is none of these.
Slope of a Segment
Slope = change in y/change in x
Given:
W(-1, 2), X(-5, 7), Y(-1, -2), and Z (3, -7)
a. Slope of WZ and slope of WX:
Slope of WZ = (-7 - 2)/(3 -(-1)) = -2.25
Slope of WX = (7 - 2)/(-1 -(-1)) = 5
b. Use distance formula, [tex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex], to find WZ and WX:
[tex]WZ = \sqrt{(3 - (-1))^2 + (-7 - 2)^2}\\\\\mathbf{WZ = \sqrt{97} }[/tex]
[tex]WX = \sqrt{(-5 -(-1))^2 + (7 - 2)^2}\\\\\mathbf{WX = \sqrt{41} }[/tex]
c. The quadrilateral WXYZ have adjacent sides that are not perpendicular to each other and have different slopes and different lengths, so therefore, WXYZ is not a rectangle, rhombus, nor a square. We can conclude that: D. WXYZ is none of these.
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