Question

for 3 - 8, use the figures below.

1. what are the coordinates of each point after quadrilateral rstu is rotated 90° about the origin? lesson 6 - 1
2. what are the coordinates of each point after quadrilateral mnpq is translated 2 units right and 5 units down? lesson 6 - 1
3. what are the coordinates of each point after quadrilateral mnpq is reflected across the y - axis and then translated 3 units left?
4. which series of transformations maps quadrilateral mnpq onto quadrilateral rstu? lesson 6 - 4

a reflection across the x - axis, translation 4 units down
b reflection across the y - axis, translation 4 units down
c rotation 180° about the origin, and then reflection across the x - axis
d rotation 180° about the origin, and then reflection across the y - axis

1. is quadrilateral mnpq congruent to quadrilateral rstu? explain. lesson 6 - 5

Explanation:

Step1: Recall rotation rules

For a 90 - degree counter - clockwise rotation about the origin, the rule is $(x,y)\to(-y,x)$.

Step2: Recall translation rules

For a translation $a$ units right and $b$ units down, the rule is $(x,y)\to(x + a,y - b)$. For a translation $a$ units left, the rule is $(x,y)\to(x - a,y)$.

Step3: Recall reflection rules

Reflection across the $x$ - axis: $(x,y)\to(x,-y)$; reflection across the $y$ - axis: $(x,y)\to(-x,y)$.

Step4: Analyze congruence

Two figures are congruent if one can be mapped onto the other by a sequence of rigid motions (translations, rotations, reflections).

a. Let the coordinates of the vertices of $RSTU$ be $(x,y)$. After a 90 - degree counter - clockwise rotation about the origin, the new coordinates $(x',y')$ are given by $(x',y')=(-y,x)$.
b. Let the coordinates of the vertices of $MNPQ$ be $(x,y)$. After translating 2 units right and 5 units down, the new coordinates are $(x + 2,y - 5)$.
c. Let the coordinates of the vertices of $MNPQ$ be $(x,y)$. After reflecting across the $x$ - axis, the coordinates become $(x,-y)$. Then after translating 3 units left, the new coordinates are $(x-3,-y)$.
d. Analyze each transformation sequence:

• For a reflection across the $x$ - axis and translation 4 units down: If the original point is $(x,y)$, after reflection across the $x$ - axis it is $(x,-y)$ and after translation 4 units down it is $(x,-y - 4)$.
• For a reflection across the $y$ - axis and translation 4 units down: If the original point is $(x,y)$, after reflection across the $y$ - axis it is $(-x,y)$ and after translation 4 units down it is $(-x,y - 4)$.
• For a rotation of 180 degrees about the origin and then reflection across the $x$ - axis: If the original point is $(x,y)$, after a 180 - degree rotation about the origin it is $(-x,-y)$, and after reflection across the $x$ - axis it is $(-x,y)$.
• For a rotation of 180 degrees about the origin and then reflection across the $y$ - axis: If the original point is $(x,y)$, after a 180 - degree rotation about the origin it is $(-x,-y)$, and after reflection across the $y$ - axis it is $(x,-y)$.
• By observing the orientation and position of $MNPQ$ and $RSTU$, we can see that a rotation of 180 degrees about the origin and then reflection across the $y$ - axis maps $MNPQ$ onto $RSTU$.

e. Since we can map $MNPQ$ onto $RSTU$ by a sequence of rigid motions (a rotation and a reflection), quadrilateral $MNPQ$ is congruent to quadrilateral $RSTU$ because rigid motions preserve shape and size.

Answer:

a. Use $(x,y)\to(-y,x)$ for each vertex of $RSTU$.
b. Use $(x,y)\to(x + 2,y - 5)$ for each vertex of $MNPQ$.
c. Use $(x,y)\to(x,-y)$ first and then $(x,-y)\to(x - 3,-y)$ for each vertex of $MNPQ$.
d. D. Rotation 180° about the origin, and then reflection across the $y$ - axis.
e. Yes, because there is a sequence of rigid motions that map $MNPQ$ onto $RSTU$.