Question

1. reason if △jkl≅△rst, give the coordinates for possible vertices of △rst. justify your answer by describing a composition of rigid motions that maps △jkl to △rst.
2. error analysis yuki says that if all lines are congruent, then all line - segments must be congruent. is yuki correct? explain.
3. mathematical connections square jklm maps to square rstu by a translation 1 unit right and 5 units up, followed by a translation 6 units left and 4 units up. what is the area of rstu?
4. higher order thinking are $overrightarrow{ab}$ and $overrightarrow{cd}$ congruent? if so, describe a composition of rigid motions that maps any ray to any other ray. if not, explain. are any two rays congruent? explain.
5. given $r_m(△pqr)=△pqr$, do △pqr and △pqr have equal perimeters? explain.
6. given wxyz≅wtuv, describe a composition of rigid motions that maps wxyz to wtuv.
7. are abcd and efgh congruent? if so, describe a composition of rigid motions that maps abcd to efgh. if not, explain.
8. which objects are congruent? for any congruent objects, describe a composition of rigid motions that maps the pre - image to the image.

Explanation:

10.

Step1: Recall congruent - triangle property

Congruent triangles have the same shape and size. Rigid motions (translations, rotations, reflections) map one congruent triangle to another.
Let's assume a translation. If $\triangle{JKL}\cong\triangle{RST}$, and we use a translation $(x,y)\to(x + a,y + b)$.
For example, if we translate $\triangle{JKL}$ 3 units to the right and 2 units up. If $J(-6,3)$, $K(-4,7)$, $L(-2,3)$, then $R(- 6+3,3 + 2)=R(-3,5)$, $S(-4 + 3,7+2)=S(-1,9)$, $T(-2+3,3 + 2)=T(1,5)$. The translation $(x,y)\to(x + 3,y + 2)$ is a composition of rigid - motions that maps $\triangle{JKL}$ to $\triangle{RST}$.

11.

Step1: Understand the concepts of lines and line - segments

A line extends infinitely in both directions, while a line - segment has two endpoints. Just because two lines are congruent (in the sense of being parallel and having the same 'direction' in a non - measurement way for lines), line - segments on them are not necessarily congruent. For example, a line can have a short line - segment and a long line - segment on it. So Yuki is incorrect.

12.

Step1: Analyze the effect of translations on area

Translations are rigid motions, and rigid motions preserve the shape and size of a figure. The area of a square $JKLM$ with side length $s = 12$ cm is $A=s^{2}$.

Step2: Calculate the area

$A = 12^{2}=144$ $cm^{2}$. Since translations preserve area, the area of $RSTU$ is also $144$ $cm^{2}$.

14.

Answer:

Step1: Check for congruence

Objects $A$ and $C$ are congruent. We can first perform a rotation to align the orientation of object $A$ with object $C$ and then perform a translation to map object $A$ onto object $C$.

There is no single answer for all these questions as they are multiple sub - questions. But for each question:

1. Coordinates of $R$, $S$, $T$ depend on the chosen rigid motion (e.g., if translation $(x,y)\to(x + 3,y + 2)$ and $J(-6,3)$, $K(-4,7)$, $L(-2,3)$, then $R(-3,5)$, $S(-1,9)$, $T(1,5)$)
2. Answer: Yuki is incorrect.
3. Answer: The area of $RSTU$ is $144$ $cm^{2}$.
4. Answer: $\triangle{P'Q'R'}$ and $\triangle{PQR}$ have equal perimeters because reflection is a rigid motion that preserves side - lengths.
5. Answer: A rotation about point $W$ followed by a translation can map $WXYZ$ to $WTUV$.
6. Answer: $ABCD$ and $EFGH$ are not congruent.
7. Answer: Objects $A$ and $C$ are congruent. A rotation followed by a translation can map $A$ to $C$.