Yes, the adjacent sides are perpendicular, and the opposite sides are parallel. (as correctly given by Option C)
How to find if a given quadrilateral is a rectangle?
A rectangle is a quadrilateral whose opposite sides are parallel to each other, and adjacent sides are perpendicular to each other.
What is the slope of a line which passes through points ( p,q) and (x,y)?
Its slope would be:
[tex]m = \dfrac{y-q}{x-p}[/tex]
Slope of parallel lines are same. Slopes of perpendicular lines are negative reciprocal of each other.
Thus, as, the adjacent sides are perpendicular, and the opposite sides are parallel. (as correctly given by Option C)
The quadrilateral in consideration is ABCD
Evaluating slopes of all 4 sides:
Case 1: Slope of AB
- Coordinates of A is (-2,2)
- Coordinates of B is (1,3)
Thus, we get:
[tex]m_{AB} = \dfrac{3-2}{1 - (-2)} = \dfrac{1}{3}[/tex]
Case 2: Slope of BC
- Coordinates of B is (1,3)
- Coordinates of C is (2,0)
Thus, we get:
[tex]m_{BC} = \dfrac{0-3}{2-1} = -3[/tex]
Case 3: Slope of DC
- Coordinates of D is (-1,-1)
- Coordinates of C is (2,0)
Thus, we get:
[tex]m_{DC} = \dfrac{0-(-1)}{2 - (-1)} = \dfrac{1}{3}[/tex]
Case 4: Slope of AD
- Coordinates of A is (-2,2)
- Coordinates of D is (-1,-1)
Thus, we get:
[tex]m_{AD} = \dfrac{-1 - 2}{-1 -(-2)} = -3[/tex]
Thus, slopes of opposite sides are equal, so they are parallel.
Slopes of adjacent sides are negative reciprocal of each other. Thus, they're perpendicular.
(example, if we take adjacent sides AD and DC, then as AD's slope is -3, the negative reciprocal of -3 is reciprocal of -(-3) which is 1/(-(-3)) = 1/3 and this is the slope of DC (and so as of AB which is also adjacent to AD, thus, AD is perpendicular to its adjacent sides DC and AB)
Thus, yes, the adjacent sides are perpendicular, and the opposite sides are parallel. (as correctly given by Option C)
Learn more about slopes here:
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