Question

1. suppose △nop is rotated and then translated to form △nop. select all of the statements that are true.

2.4 in.
27°
a. ∠n is an acute angle.
b. m∠o = 63
c. length no = 2.4 in
d. ∠n≅∠n
e. po≅op

1. point t is at (-6, 4). what are the coordinates of point t after a reflection across x = 0 and then y = 0? (-6), (4)
2. gh is reflected across the line y = 2. select all that apply.

a. g is plotted at (3, 1).
b. g is plotted at (-3, -1).
c. h and h are in the same quadrant.
d. gh≅gh
e. g and g are in the same quadrant.

1. which transformation or sequence of transformations maps △abc to △abc?

a a translation 5 units left, then a reflection across y = 0
b a translation 5 units left, then a rotation 90° about the origin
c a reflection across y = 0
d a rotation 90° about the origin

1. point p(-3, 2) is the image of point p(3, 8) under a translation. select the image of (0, -6) under the same translation.

a (-3, -4)
b (0, 6)
c (0, 10)
d (-6, -12)

1. given point a is on line m, suppose a is reflected across line m. which of the following is true?

a aa is the bisector of line m.
b line m is the perpendicular bisector of aa.
c line m is parallel to aa.
d a = a

Explanation:

Step1: Recall properties of rigid - motions

Rigid - motions (rotations, translations, reflections) preserve angle measures and side lengths. So, for a triangle $\triangle NOP$ rotated and translated to form $\triangle N'O'P'$, corresponding angles are congruent and corresponding sides are congruent. $\angle N\cong\angle N'$, $\overline{PO}\cong\overline{O'P'}$, and if $\angle O = 63^{\circ}$, then $m\angle O'=63^{\circ}$. Also, if $\overline{NO}=2.4$ in, its length is preserved. And since rotation and translation are rigid - motions, $\angle N'$ has the same measure as $\angle N$. If $\angle N$ is acute in $\triangle NOP$, $\angle N'$ is acute.
For point $T(-6,4)$:

Step2: Reflection across $x = 0$

The rule for reflecting a point $(x,y)$ across the line $x = 0$ (the $y$ - axis) is $(-x,y)$. So, $T(-6,4)$ becomes $T_1(6,4)$.

Step3: Reflection across $y = 0$

The rule for reflecting a point $(x,y)$ across the line $y = 0$ (the $x$ - axis) is $(x,-y)$. So, $T_1(6,4)$ becomes $T'=(6, - 4)$.
For the reflection of $\overline{GH}$ across $y = 2$:

Step4: Analyze reflection rules

The distance between a point and the line of reflection is preserved. Let's assume $G(-3,4)$ and $H(1,2)$. The distance from $G$ to $y = 2$ is $4 - 2=2$ units. The new $y$ - coordinate of $G'$ is $2-2 = 0$. The $x$ - coordinate of $G'$ is the same as that of $G$ since the reflection is across a horizontal line. So, $G'(-3,0)$. The distance from $H$ to $y = 2$ is $2 - 2=0$. So, $H'$ has the same coordinates as $H$. And $\overline{GH}\cong\overline{G'H'}$ because reflection is a rigid - motion.
For the transformation of $\triangle ABC$ to $\triangle A'B'C'$:

Step5: Analyze transformation rules

The coordinates of $A(5,5)$ change to $A'(0, - 5)$. First, a translation 5 units left changes $(x,y)$ to $(x - 5,y)$. So, $A(5,5)$ becomes $(0,5)$. Then a reflection across $y = 0$ (the $x$ - axis) changes $(x,y)$ to $(x,-y)$. So, $(0,5)$ becomes $(0,-5)$.
For the translation of point $P(3,8)$ to $P'(-3,2)$:

Step6: Find the translation rule

The $x$ - coordinate changes from $3$ to $-3$, so $\Delta x=-3 - 3=-6$. The $y$ - coordinate changes from $8$ to $2$, so $\Delta y=2 - 8=-6$. For the point $(0,-6)$, the new $x$ - coordinate is $0-6=-6$ and the new $y$ - coordinate is $-6-6=-12$.
For the reflection of point $A$ across line $m$:

Step7: Recall reflection properties

The line of reflection is the perpendicular bisector of the segment joining a point and its image. So, line $m$ is the perpendicular bisector of $\overline{AA'}$.

Answer:

1. A. $\angle N'$ is an acute angle. (True if $\angle N$ is acute in the original triangle)

B. $m\angle O' = 63$ (True if $m\angle O = 63$ in the original triangle)
C. length $\overline{NO}=2.4$ in (True as length is preserved in rigid - motions)
D. $\angle N\cong\angle N'$ (True as rigid - motions preserve angle measures)
E. $\overline{PO}\cong\overline{O'P'}$ (True as rigid - motions preserve side lengths)

1. $(6,-4)$
2. D. $\overline{GH}\cong\overline{G'H'}$
3. A. a translation 5 units left, then a reflection across $y = 0$
4. D. $(-6,-12)$
5. B. Line $m$ is the perpendicular bisector of $\overline{AA'}$