Question

line segment $overline{ef}$ has endpoints at $(-4, -14)$ and $(-3, 7)$. line segment $overline{uv}$ has endpoints at $(-7, 5)$ and $(14, 6)$. are the line segments congruent or not congruent? congruent with a length of 340 units congruent with a length of 200 units congruent with a length of $sqrt{442}$ units

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate length of $\overline{EF}$

For endpoints $E(-4,-14)$ and $F(-3,7)$:
$d_{EF}=\sqrt{(-3-(-4))^2+(7 - (-14))^2}=\sqrt{(1)^2+(21)^2}=\sqrt{1 + 441}=\sqrt{442}$.

Step3: Calculate length of $\overline{UV}$

For endpoints $U(-7,5)$ and $V(14,6)$:
$d_{UV}=\sqrt{(14-(-7))^2+(6 - 5)^2}=\sqrt{(21)^2+(1)^2}=\sqrt{441+1}=\sqrt{442}$.

Step4: Compare lengths

Since $d_{EF}=\sqrt{442}$ and $d_{UV}=\sqrt{442}$, the line - segments are congruent with a length of $\sqrt{442}$ units.

Answer:

Congruent with a length of $\sqrt{442}$ units