Question

rotations notes 1
previously, we rotated points counter - clockwise and about the origin.
rotation rules from last week:
90°: (x, y)→(-y, x)
180°: (x, y)→(-x, -y)
270°: (x, y)→(y, -x)
rotations notes 2
why dont we have a rule for rotations of 360°?
it stays the same.
when combined with translations, these rules can also be used to rotate figures about points that are not the origin.
guided practice
rotate the point (-5, 2) 180° about the point (1, 4).
step 1: map the center of rotation to the origin.
the center of rotation is (1, 4)
to map (1, 4)→(0, 0), the rule is:
(x, y)→(x - 1, y - 4)
translate the point (-5, 2) using the rule (x, y)→(x - 1, y - 4).
(-5 - 1, 2 - 4)=(-6, -2)
step 2: rotate the point (-6, -2) 180°
(x, y)→(-x, -y)
(-6, -2)→(6, 2)
step 3: map the center of rotation back to its original position.
this will be the ____________ of the vector in step 1.
vector from step 1: (x - 1, y - 4)
the opposite of this vector is (x + 1, y + 4)
step 4: translate the point from step 2 along the same vector.
translate (6, 2) along (x + 1, y + 4)
(6, 2)→(, )
our final answer is (7, 6)

Explanation:

Step1: Identify the rotation - center mapping rule

To map the center of rotation $(1,4)$ to the origin $(0,0)$, the translation rule is $(x,y)\to(x - 1,y - 4)$.

Step2: Translate the point to be rotated

For the point $(-5,2)$, using the rule $(x,y)\to(x - 1,y - 4)$, we get $(-5-1,2 - 4)=(-6,-2)$.

Step3: Apply the rotation rule

The rotation rule for $180^{\circ}$ about the origin is $(x,y)\to(-x,-y)$. For the point $(-6,-2)$ after translation, it becomes $(6,2)$.

Step4: Reverse - translate the rotated point

Since we first translated the center of rotation to the origin, we now need to reverse the translation. The reverse of the vector $(x - 1,y - 4)$ is $(x + 1,y + 4)$. For the point $(6,2)$, applying the rule $(x,y)\to(x + 1,y + 4)$ gives $(6 + 1,2+4)=(7,6)$.

Answer:

$(7,6)$