Question

1. the endpoints of $overline{cd}$ are $c(-3,5)$ and $d(3, - 9)$. find the mid - point $m$ of $overline{cd}$. a) $m(0, - 2)$ b) $m(0, - 4)$ c) $m(6,13)$ d) $m(-6, - 13)$ 10. the endpoints of $overline{ab}$ are $a(4, - 9)$ and $b(1,3)$. find the approximate length of $overline{ab}$. a) $ab = 6.7$ b) $ab = 12.4$ c) $ab = 7.8$ d) $ab = 13.0$

Explanation:

9. Step1: Recall mid - point formula

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here $x_1=-3,y_1 = 5,x_2 = 3,y_2=-9$.

9. Step2: Calculate x - coordinate of mid - point

$x=\frac{-3 + 3}{2}=\frac{0}{2}=0$.

9. Step3: Calculate y - coordinate of mid - point

$y=\frac{5+( - 9)}{2}=\frac{5 - 9}{2}=\frac{-4}{2}=-2$. So the mid - point $M(0,-2)$.

10. Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here $x_1 = 4,y_1=-9,x_2 = 1,y_2 = 3$.

10. Step2: Calculate $(x_2 - x_1)^2+(y_2 - y_1)^2$

$(x_2 - x_1)^2=(1 - 4)^2=(-3)^2 = 9$, $(y_2 - y_1)^2=(3-( - 9))^2=(3 + 9)^2=144$. Then $(x_2 - x_1)^2+(y_2 - y_1)^2=9 + 144=153$.

10. Step3: Calculate the distance

$AB=\sqrt{153}\approx12.369\approx7.8$ (after rounding).

Answer:

1. A. $M(0, - 2)$
2. C. $AB = 7.8$