Question

directions: use the distance and slope formula to determine whether $overline{ab}$ and $overline{cd}$ are equivalent.

1. $overline{ab}$ with $a(4, 8)$ and $b(6, -9)$; $overline{cd}$ with $c(-3, 11)$ and $d(-1, -6)$

$ab=sqrt{4 + (-17)^2}=sqrt{293}=17.1$

1. $overline{ab}$ with $a(1, 2)$ and $b(-1, -5)$; $overline{cd}$ with $c(-8, -1)$ and $d(-6, -8)$

Explanation:

For Problem 11:

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

Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$$\overline{AB}=\sqrt{(6-4)^2+(-9-8)^2}=\sqrt{2^2+(-17)^2}=\sqrt{4+289}=\sqrt{293}$$

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

$$\overline{CD}=\sqrt{(-1-(-3))^2+(-6-11)^2}=\sqrt{2^2+(-17)^2}=\sqrt{4+289}=\sqrt{293}$$

Step3: Calculate slope of $\overline{AB}$

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$$m_{AB}=\frac{-9-8}{6-4}=\frac{-17}{2}$$

Step4: Calculate slope of $\overline{CD}$

$$m_{CD}=\frac{-6-11}{-1-(-3)}=\frac{-17}{2}$$

Step5: Compare length and slope

Segments are equivalent if same length and slope.

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

$$\overline{AB}=\sqrt{(-1-1)^2+(-5-2)^2}=\sqrt{(-2)^2+(-7)^2}=\sqrt{4+49}=\sqrt{53}$$

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

$$\overline{CD}=\sqrt{(-6-(-8))^2+(-8-(-1))^2}=\sqrt{2^2+(-7)^2}=\sqrt{4+49}=\sqrt{53}$$

Step3: Calculate slope of $\overline{AB}$

$$m_{AB}=\frac{-5-2}{-1-1}=\frac{-7}{-2}=\frac{7}{2}$$

Step4: Calculate slope of $\overline{CD}$

$$m_{CD}=\frac{-8-(-1)}{-6-(-8)}=\frac{-7}{2}$$

Step5: Compare length and slope

Slopes are not equal, so segments are not equivalent.

Answer:

$\overline{AB}$ and $\overline{CD}$ are equivalent.

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For Problem 12: