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Choose all that define a rectangle in the coordinate plane by the four given points.

A(−3, 0), B(3, 2), C(4, −1), D(−2, −3)
A(−2, −5), B(−2, 1), C(3, 2), D(1, −4)
A(−1, −1), B(2, 3), C(10, −3), D(7, −7)
A(−3, −4), B(−1, 2), C(2, 1), D(0, −5)

Respuesta :

maggiesnow4

Answer:

all but the second option

Step-by-step explanation:

all of them except the second one

In the segments joining the points that define a rectangle are; Opposite

segments are equal and adjacent segments are perpendicular.

Responses:

The rectangles are;

  • A(-3, 0), B(3, 2), C(4, -1), D(-2, -3)
  • A(-1, -1), B(2, 3), C(10, -3), D(7, -7)
  • A(-3, -4), B(-1, 2), C(2, 1), D(0, -5)

How can the points that define a rectangle be found?

A rectangle is a quadrilateral that have opposite sides equal and

adjacent sides perpendicular.

Therefore;

First option, we have;

Length of AB = √((3 - (-3))² + (2 - 0)²) = 2·√10

Length of BC = √((3 - 4)² + (2 - (-1))²) = √10

Length of CD = √((4 - (-2))² + (-1 - (-3))²) = 2·√10

Length of DA = √((-3 - (-2))² + (0 - (-3))²) = √10

Slope of side AB = [tex]\dfrac{2}{6}[/tex] = [tex]\dfrac{1}{3}[/tex]

Slope of side BC = [tex]\dfrac{3}{-1}[/tex] = -3

Slope of side CD = [tex]\dfrac{2}{6}[/tex] = [tex]\mathbf{\dfrac{1}{3}}[/tex]

Slope of side DA = [tex]\dfrac{3}{-1}[/tex] = -3

The length of the opposite sides are equal.

The slope of the adjacent sides are the negative inverse of each other, therefore, the adjacent sides are perpendicular.

  • The quadrilateral [tex]\underline{A(-3, \, 0), B(3, \, 2), C(4, \, -1), D(-2, \, -3)}[/tex] is a rectangle.

Second option, we have;

Length of AB = √(0)² + (-6)²) = 6

Length of BC = √(5² + 1²) = √(26)

Length of CD = √((2)² + (2 - (-4))²) = 2·√10

The opposite sides of the quadrilateral in the second option are not equal

Third option, we have;

Length of AB = √(3² + 4²) = 5

Length of BC = √((10 - 2)² + (-3 - 3)²) = 10

Length of CD = √(3² + 4²) = 5

Length of DA = √((7 - (-1))² + (-7 - (-1))²) = 10

Slope of side AB = [tex]\mathbf{\dfrac{4}{3}}[/tex]

Slope of side BC = [tex]\dfrac{-6}{8} = \mathbf{-\dfrac{3}{4}}[/tex]

Slope of side CD = [tex]\dfrac{-4}{-3} = \dfrac{4}{3}[/tex]

Slope of side DA = [tex]\dfrac{-6}{8} = -\dfrac{3}{4}[/tex]

The length of the opposite sides are equal.

The slope of the adjacent sides are the negative inverse of each other,

therefore, the adjacent sides are perpendicular.

  • The quadrilateral [tex]\underline{A(-1, \, -1), B(2, \, 3), C(10, \, -3), D(7, \, -7)}[/tex] is a rectangle.

Fourth option, we have;

Length of AB = √(2² + 6²) = 2·√10

Length of BC = √(3² + (-1)²) = √10

Length of CD = √(2² + 6²) = 2·√10

Length of DA = √((0 - (-3))² + (-5 - (-4))²) = √10

Slope of side AB = [tex]\dfrac{6}{2}[/tex] = 3

Slope of side BC = [tex]\dfrac{1}{-3} = \mathbf{-\dfrac{1}{3}}[/tex]

Slope of side CD = [tex]\dfrac{6}{2}[/tex] = 3

Slope of side DA = [tex]\dfrac{1}{-3} = -\dfrac{1}{3}[/tex]

The length of the opposite sides are equal.

The slope of the adjacent sides are the negative inverse of each other, therefore, the adjacent sides are perpendicular.

  • The quadrilateral,  [tex]\underline{A(-3, \, -4), B(-1, \, 2), C(2, \, 1), D(0, \, -5)}[/tex] is a rectangle

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