Question

the graph shows triangles opq and vwx. is opq congruent to vwx? justify your answer. yes, because a translation right 13 units and up 1 unit maps opq onto vwx. yes, because a reflection across the y - axis maps opq onto vwx. no, because $overline{op}$ and $overline{vw}$ do not have the same length. no, because $overline{pq}$ and $overline{wx}$ do not have the same length.

Explanation:

Step1: Recall congruence criteria

Two triangles are congruent if all corresponding sides and angles are equal. One - way to check is by comparing side - lengths.

Step2: Use the 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 example, if $O(-8,-4)$ and $P(-6,-3)$, then $OP=\sqrt{(-6+8)^2+(-3 + 4)^2}=\sqrt{4 + 1}=\sqrt{5}$. If we assume $V(5,-3)$ and $W(7,-2)$, then $VW=\sqrt{(7 - 5)^2+(-2+3)^2}=\sqrt{4 + 1}=\sqrt{5}$.

Step3: Analyze other sides

Let's assume $Q(-7,5)$ and $X(6,6)$. The length of $PQ=\sqrt{(-7 + 6)^2+(5 + 3)^2}=\sqrt{1+64}=\sqrt{65}$. The length of $WX=\sqrt{(6 - 7)^2+(6 + 2)^2}=\sqrt{1 + 64}=\sqrt{65}$.

Step4: Check for congruence

Since all corresponding sides of $\triangle OPQ$ and $\triangle VWX$ have the same lengths (by calculating distances between vertices using the distance formula), we can also check the transformation. If we take a point $O(-8,-4)$ and apply a translation right 13 units and up 1 unit: $(-8+13,-4 + 1)=(5,-3)$ which is the coordinates of $V$. For point $P(-6,-3)$, $(-6+13,-3 + 1)=(7,-2)$ which is the coordinates of $W$. For point $Q(-7,5)$, $(-7+13,5 + 1)=(6,6)$ which is the coordinates of $X$.

Answer:

Yes, because a translation right 13 units and up 1 unit maps $OPQ$ onto $VWX$.