2. graph $ \triangle p q r $ with vertices $ p ( - 7, - 4 ), q ( - 4,0 ) $ and $ r ( - 1, - 4 ) $ on the graph provided.
a. rotate $ \triangle p q r 180 ^ { \circ } $ about the origin to construct $ \triangle x y z $.
b. list the side lengths for all segments.
$ p q = $____ $ x y = $____
$ q r = $____ $ y z = $____
$ r p = $____ $ z x = $____
c. mark the sides that are congruent on the graph. list all sides that are congruent.
$ \overline { p q } \cong $____
$ \overline { q r } \cong $____
$ \overline { r p } \cong $____
e. verify angles are congruent using a protractor. mark the angles that are congruent on the graph. list all angles that are congruent.
$ \angle p \cong $____
$ \angle q \cong $____
$ \angle r \cong $____
e. if all sides and angles of two triangles are congruent, then the two triangles are congruent. therefore,
$ \triangle $____ $ \cong \triangle $____.
for problems 3 - 5, list all congruent parts based on the given congruence statement.
3. $ \triangle j k l \cong \triangle m n o $
$ \overline { j k } \cong $____
$ \overline { k l } \cong $____
$ \overline { l j } \cong $____
$ \angle j \cong $____
$ \angle k \cong $____
$ \angle l \cong $____
4. $ \triangle c a r \cong \triangle b u s $
$ \overline { c a } \cong $____
$ \overline { a r } \cong $____
$ \overline { r c } \cong $____
$ \angle c \cong $____
$ \angle a \cong $____
$ \angle r \cong $____
5. $ \triangle a r m \cong \triangle l e g $
$ \overline { a r } \cong $____
$ \overline { r m } \cong $____
$ \overline { m a } \cong $____
$ \angle a \cong $____
$ \angle r \cong $____
$ \angle m \cong $____

Question

Accepted Answer

# Explanation:
## Step1: Distance formula
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $PQ$ with $P(-7,-4)$ and $Q(-4,0)$:
$$
\begin{align*}
PQ&=\sqrt{(-4+7)^2+(0 + 4)^2}\
&=\sqrt{3^2+4^2}\
&=\sqrt{9 + 16}\
&=\sqrt{25}\
&=5
\end{align*}
$$
Since rotation is a rigid transformation, $XY = PQ$. So $XY=5$.
## Step2: For $QR$ with $Q(-4,0)$ and $R(-1,-4)$
$$
\begin{align*}
QR&=\sqrt{(-1 + 4)^2+(-4-0)^2}\
&=\sqrt{3^2+(-4)^2}\
&=\sqrt{9+16}\
&=\sqrt{25}\
&=5
\end{align*}
$$
Since rotation is a rigid transformation, $YZ = QR$. So $YZ = 5$.
## Step3: For $RP$ with $R(-1,-4)$ and $P(-7,-4)$
$$
\begin{align*}
RP&=\sqrt{(-7 + 1)^2+(-4+4)^2}\
&=\sqrt{(-6)^2+0^2}\
&=\sqrt{36}\
&=6
\end{align*}
$$
Since rotation is a rigid transformation, $ZX=RP$. So $ZX = 6$.
## Step4: Congruent sides
Since $\triangle PQR$ is rotated $180^{\circ}$ about the origin to get $\triangle XYZ$ (a rigid transformation), $\overline{PQ}\cong\overline{XY}$, $\overline{QR}\cong\overline{YZ}$, $\overline{RP}\cong\overline{ZX}$.
## Step5: Congruent angles
Since rotation is a rigid transformation (preserves angles), $\angle P\cong\angle X$, $\angle Q\cong\angle Y$, $\angle R\cong\angle Z$. And $\triangle PQR\cong\triangle XYZ$ (by SSS or ASA or AAS  since all corresponding sides and angles are congruent).
## Step6: Congruent parts for $\triangle JKL\cong\triangle MNO$
Corresponding parts: $\overline{JK}\cong\overline{MN}$, $\overline{KL}\cong\overline{NO}$, $\overline{LJ}\cong\overline{OM}$, $\angle J\cong\angle M$, $\angle K\cong\angle N$, $\angle L\cong\angle O$
## Step7: Congruent parts for $\triangle CAR\cong\triangle BUS$
Corresponding parts: $\overline{CA}\cong\overline{BU}$, $\overline{AR}\cong\overline{US}$, $\overline{RC}\cong\overline{SB}$, $\angle C\cong\angle B$, $\angle A\cong\angle U$, $\angle R\cong\angle S$
## Step8: Congruent parts for $\triangle ARM\cong\triangle LEG$
Corresponding parts: $\overline{AR}\cong\overline{LE}$, $\overline{RM}\cong\overline{EG}$, $\overline{MA}\cong\overline{GL}$, $\angle A\cong\angle L$, $\angle R\cong\angle E$, $\angle M\cong\angle G$

# Answer:
- $PQ = 5$, $XY=5$, $QR = 5$, $YZ = 5$, $RP=6$, $ZX = 6$
- $\overline{PQ}\cong\overline{XY}$, $\overline{QR}\cong\overline{YZ}$, $\overline{RP}\cong\overline{ZX}$
- $\angle P\cong\angle X$, $\angle Q\cong\angle Y$, $\angle R\cong\angle Z$, $\triangle PQR\cong\triangle XYZ$
- For $\triangle JKL\cong\triangle MNO$: $\overline{JK}\cong\overline{MN}$, $\overline{KL}\cong\overline{NO}$, $\overline{LJ}\cong\overline{OM}$, $\angle J\cong\angle M$, $\angle K\cong\angle N$, $\angle L\cong\angle O$
- For $\triangle CAR\cong\triangle BUS$: $\overline{CA}\cong\overline{BU}$, $\overline{AR}\cong\overline{US}$, $\overline{RC}\cong\overline{SB}$, $\angle C\cong\angle B$, $\angle A\cong\angle U$, $\angle R\cong\angle S$
- For $\triangle ARM\cong\triangle LEG$: $\overline{AR}\cong\overline{LE}$, $\overline{RM}\cong\overline{EG}$, $\overline{MA}\cong\overline{GL}$, $\angle A\cong\angle L$, $\angle R\cong\angle E$, $\angle M\cong\angle G$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Distance formula

Step2: For \(QR\) with \(Q(-4,0)\) and \(R(-1,-4)\)

Step3: For \(RP\) with \(R(-1,-4)\) and \(P(-7,-4)\)

Step4: Congruent sides

Step5: Congruent angles

Step6: Congruent parts for \(\triangle JKL\cong\triangle MNO\)

Step7: Congruent parts for \(\triangle CAR\cong\triangle BUS\)

Step8: Congruent parts for \(\triangle ARM\cong\triangle LEG\)

Answer: