Question

List the coordinates of the image after a rotation of the figure below of 180° about the origin. 20 points

(__ , __)
(__ , __)
(__ , __)
(__ , __)
Find the measures of the segment and its image:

AD = √___, A'D' = √___

Answer 1

Answer: The co-ordinates of the image quadrilateral areA'(-3, -1), B'(-5, -4), C'(-7, -2) and D'(-4, 0). The length of the segments AD and A'D' are both √2 units.

Step-by-step explanation: We are given to list the co-ordinates of the image after a rotation of the figure below of 180° about the origin.

From the figure, we note that

the co-ordinates of the vertices of quadrilateral ABCD are A(3, 1), B(5, 4), C(7, 2) and D(4, 0).

We know that

the co-ordinates of the image of a point (x, y) after rotation of 180° about the origin is given by (-x, -y).

Therefore, after rotating the quadrilateral ABCD through an angle of 180° about the origin, then the co-ordinates of the image A'B'C'D' are

A(3, 1) ⇒ A'(-3, -1),

B(5, 4) ⇒ B'(-5, -4),

C(7, 2) ⇒ C'(-7, -2)

and

D(4, 0) ⇒ D'(-4, 0).

And, the lengths of the line segments AD and A'D' can be calculated using distance formula as follows :

`AD=√((4-3)^2+(0-1)^2)=√(1+1)=√(2)~\textup{units},\\\\A'D'=√((-4+3)^2+(0+1)^2)=√(1+1)=\sqrt 2~\textup{units}.`

Thus, the co-ordinates of the image quadrilateral are A'(-3, -1), B'(-5, -4), C'(-7, -2) and D'(-4, 0). The length of the segments AD and A'D' are both √2 units.

Answer 2

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

AD = sqrt(2)

A'D' = sqrt(2)