the graph shows triangles def and klm.
is def congruent to klm? justify your answer.
yes, because a rotation 90° counterclockwise around the origin maps def onto klm.
yes, because a rotation 90° counterclockwise around d followed by a translation right 9 units and down 5 units maps def onto klm.
no, because ∠d and ∠k do not have the same measure.
no, because $\overline{de}$ and $\overline{kl}$ do not have the same length.

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Accepted Answer

# Brief Explanations:
To determine if $\triangle DEF$ and $\triangle KLM$ are congruent, we analyze the side lengths. First, find the coordinates of the vertices:
- $D(-8, -2)$, $E(-5, -7)$, $F(-3, -6)$
- $K(1, -8)$, $L(6, -5)$, $M(5, -3)$

Calculate the length of $\overline{DE}$ using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$:
$$
\begin{align*}
DE&=\sqrt{(-5 - (-8))^2 + (-7 - (-2))^2}\
&=\sqrt{(3)^2 + (-5)^2}\
&=\sqrt{9 + 25}\
&=\sqrt{34}
\end{align*}
$$

Calculate the length of $\overline{KL}$:
$$
\begin{align*}
KL&=\sqrt{(6 - 1)^2 + (-5 - (-8))^2}\
&=\sqrt{(5)^2 + (3)^2}\
&=\sqrt{25 + 9}\
&=\sqrt{34}
\end{align*}
$$

So, $\overline{DE}$ and $\overline{KL}$ have the same length. Now, check the transformation options:
- The first option (rotation $90^\circ$ counterclockwise around the origin) would not map $D$ to $K$ correctly.
- The second option's transformation is more complex and unnecessary since we can check side lengths.
- The third option is incorrect because $\angle D$ and $\angle K$ should be equal if the triangles are congruent (by rotation/reflection/translation), but we can also check side lengths.
- The fourth option is incorrect because we just calculated $DE = KL=\sqrt{34}$. Wait, no—wait, let's re - evaluate. Wait, maybe I made a mistake. Wait, let's check the other sides. Wait, no, the key is: the correct answer is the one that correctly identifies congruence via transformation. Wait, actually, when we rotate $90^\circ$ counterclockwise around the origin, the transformation of a point $(x,y)$ is $(-y,x)$. Let's apply this to $D(-8, -2)$: $(2, -8)$, which is not $K(1, -8)$. So the first option is wrong. The second option: rotate $90^\circ$ counterclockwise around $D$, then translate. But actually, let's check the side lengths again. Wait, maybe the fourth option is wrong. Wait, no, let's calculate $DE$ and $KL$ again. $D(-8,-2)$, $E(-5,-7)$: the difference in $x$ is $-5 - (-8)=3$, difference in $y$ is $-7 - (-2)= - 5$. So $DE=\sqrt{3^2+(-5)^2}=\sqrt{9 + 25}=\sqrt{34}$. $K(1,-8)$, $L(6,-5)$: difference in $x$ is $6 - 1 = 5$, difference in $y$ is $-5-(-8)=3$. So $KL=\sqrt{5^2 + 3^2}=\sqrt{25 + 9}=\sqrt{34}$. So $DE = KL$. Now, what about the other sides? Let's calculate $DF$ and $KM$. $D(-8,-2)$, $F(-3,-6)$: difference in $x$ is $-3-(-8)=5$, difference in $y$ is $-6 - (-2)=-4$. $DF=\sqrt{5^2+(-4)^2}=\sqrt{25 + 16}=\sqrt{41}$. $K(1,-8)$, $M(5,-3)$: difference in $x$ is $5 - 1 = 4$, difference in $y$ is $-3-(-8)=5$. $KM=\sqrt{4^2 + 5^2}=\sqrt{16+25}=\sqrt{41}$. And $EF$ and $LM$: $E(-5,-7)$, $F(-3,-6)$: difference in $x$ is $-3-(-5)=2$, difference in $y$ is $-6-(-7)=1$. $EF=\sqrt{2^2+1^2}=\sqrt{5}$. $L(6,-5)$, $M(5,-3)$: difference in $x$ is $5 - 6=-1$, difference in $y$ is $-3-(-5)=2$. $LM=\sqrt{(-1)^2+2^2}=\sqrt{1 + 4}=\sqrt{5}$. So all sides are equal. Now, the correct transformation: a $90^\circ$ counterclockwise rotation around the origin? Wait, no, when we rotate $90^\circ$ counterclockwise around the origin, $(x,y)\to(-y,x)$. For $D(-8,-2)$, it becomes $(2,-8)$, which is not $K(1,-8)$. But if we rotate $90^\circ$ counterclockwise around a different point? Wait, the first option says "rotation $90^\circ$ counterclockwise around the origin maps $DEF$ onto $KLM$". But when we apply the rotation to $D(-8,-2)$, we get $(2,-8)$, which is not $K(1,-8)$. So the first option is wrong. The second option: "rotation $90^\circ$ counterclockwise around $D$ followed by a translation right 9 units and down 5 units". Let's try rotating $90^\circ$ counterclockwise around $D(-8,-2)$. The formula for rotating a point $(x,y)$ around $(a,b)$ by $90^\circ$ counterclockwise is $(a-(y - b),b+(x - a))$. For point $E(-5,-7)$ around $D(-8,-2)$: $x=-5,y=-7,a=-8,b=-2$. So $a-(y - b)=-8-(-7 + 2)=-8-(-5)=-3$, $b+(x - a)=-2+(-5 + 8)=-2 + 3 = 1$. So the rotated point of $E$ around $D$ is $(-3,1)$. Then translating right 9 units: $-3 + 9 = 6$, down 5 units: $1-5=-4$. Wait, that's not $L$ or $M$. Maybe my approach is wrong. Alternatively, since we know the triangles are congruent (all sides equal), the correct answer is the one that says yes, and the first option is incorrect, the second option's transformation is a bit convoluted, but actually, the mistake was in my initial side - length calculation? No, the side lengths are equal. Wait, the correct answer is: the first option is wrong, the second option is a valid transformation (even if complex), the third option is wrong (angles should be equal), the fourth option is wrong (sides are equal). Wait, no, I think I messed up. Wait, let's check the coordinates again. $D(-8,-2)$, $K(1,-8)$: the vector from $D$ to $K$ is $(9,-6)$. If we rotate $90^\circ$ counterclockwise around the origin, the vector $(x,y)$ becomes $(-y,x)$. The vector $DE$ is $(3,-5)$, rotating it $90^\circ$ counterclockwise gives $(5,3)$, which is the vector $KL=(5,3)$ (since $KL$ is from $K(1,-8)$ to $L(6,-5)$, the vector is $(5,3)$). Ah! So the rotation of $\overline{DE}$ by $90^\circ$ counterclockwise around the origin gives $\overline{KL}$, and the rotation of $\overline{DF}$ by $90^\circ$ counterclockwise around the origin gives $\overline{KM}$, and the rotation of $\overline{EF}$ by $90^\circ$ counterclockwise around the origin gives $\overline{LM}$. So the first option is correct. Because when we rotate $\triangle DEF$ $90^\circ$ counterclockwise around the origin, each side vector is rotated to match the side vectors of $\triangle KLM$, so the rotation maps $DEF$ onto $KLM$, proving they are congruent.

# Answer:
Yes, because a rotation $90^\circ$ counterclockwise around the origin maps $DEF$ onto $KLM$.

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