Question

segment ab has endpoints at (-5,7) and (-8,10). segment cd has endpoints at (1,-7) and (-3,-5). segment ef has endpoints at (0,8) and (2,12). which segments are congruent? options: ab ≅ cd, none of the segments are congruent, cd ≅ ef, ab ≅ ef

Explanation:

Step1: Calculate length of $\overline{AB}$

Use distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For $\overline{AB}$ with $(-5,7)$ and $(-8,10)$:
$d_{AB}=\sqrt{(-8 - (-5))^2 + (10 - 7)^2}=\sqrt{(-3)^2 + 3^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}$

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

For $\overline{CD}$ with $(1,-7)$ and $(-3,-5)$:
$d_{CD}=\sqrt{(-3 - 1)^2 + (-5 - (-7))^2}=\sqrt{(-4)^2 + 2^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$

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

For $\overline{EF}$ with $(0,8)$ and $(2,12)$:
$d_{EF}=\sqrt{(2 - 0)^2 + (12 - 8)^2}=\sqrt{2^2 + 4^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}$

Step4: Compare lengths

$d_{AB}=3\sqrt{2}$, $d_{CD}=2\sqrt{5}$, $d_{EF}=2\sqrt{5}$. So $d_{CD}=d_{EF}$, meaning $\overline{CD}\cong\overline{EF}$.

Answer:

$\overline{CD}\cong\overline{EF}$