Question

luis is programming an animation for a countdown timer where points flash in sequence, one at a time, around in a circle. he calculates that the coordinates of the first four points in his sequence are (6,0), (5.5,2.3), (4.2,4.2), and (2.3,5.5). he can find the rest of the coordinates by rotating the first four points 90°, 180°, and 270°. what are the coordinates that complete the sequence around in a circle? when rotated 90°, the points are (type ordered pairs. use a comma to separate answers as needed.)

Explanation:

Step1: Recall rotation rules for 90 - degree rotation

For a 90 - degree counter - clockwise rotation of a point $(x,y)$ about the origin, the new coordinates $(x',y')$ are given by $(-y,x)$.

Step2: Rotate the point $(6,0)$ 90 - degrees

For the point $(6,0)$, using the rule $(-y,x)$, we have $(-0,6)=(0,6)$.

Step3: Rotate the point $(5.5,2.3)$ 90 - degrees

For the point $(5.5,2.3)$, $x = 5.5$ and $y = 2.3$. The new point is $(-2.3,5.5)$.

Step4: Rotate the point $(4.2,4.2)$ 90 - degrees

For the point $(4.2,4.2)$, the new point is $(-4.2,4.2)$.

Step5: Rotate the point $(2.3,5.5)$ 90 - degrees

For the point $(2.3,5.5)$, the new point is $(-5.5,2.3)$.

Step6: Recall rotation rules for 180 - degree rotation

For a 180 - degree rotation of a point $(x,y)$ about the origin, the new coordinates $(x',y')$ are given by $(-x,-y)$.

Step7: Rotate the point $(6,0)$ 180 - degrees

For the point $(6,0)$, the new point is $(-6,0)$.

Step8: Rotate the point $(5.5,2.3)$ 180 - degrees

For the point $(5.5,2.3)$, the new point is $(-5.5,-2.3)$.

Step9: Rotate the point $(4.2,4.2)$ 180 - degrees

For the point $(4.2,4.2)$, the new point is $(-4.2,-4.2)$.

Step10: Rotate the point $(2.3,5.5)$ 180 - degrees

For the point $(2.3,5.5)$, the new point is $(-2.3,-5.5)$.

Step11: Recall rotation rules for 270 - degree rotation

For a 270 - degree counter - clockwise rotation of a point $(x,y)$ about the origin (which is equivalent to a 90 - degree clockwise rotation), the new coordinates $(x',y')$ are given by $(y, - x)$.

Step12: Rotate the point $(6,0)$ 270 - degrees

For the point $(6,0)$, the new point is $(0,-6)$.

Step13: Rotate the point $(5.5,2.3)$ 270 - degrees

For the point $(5.5,2.3)$, the new point is $(2.3,-5.5)$.

Step14: Rotate the point $(4.2,4.2)$ 270 - degrees

For the point $(4.2,4.2)$, the new point is $(4.2,-4.2)$.

Step15: Rotate the point $(2.3,5.5)$ 270 - degrees

For the point $(2.3,5.5)$, the new point is $(5.5,-2.3)$.

Answer:

$(0,6),(-2.3,5.5),(-4.2,4.2),(-5.5,2.3),(-6,0),(-5.5,-2.3),(-4.2,-4.2),(-2.3,-5.5),(0,-6),(2.3,-5.5),(4.2,-4.2),(5.5,-2.3)$