Question

for questions 1 to 3, choose the correct answer. 1. the mid - point of the line segment with endpoints a(-3, -3) and b(1, 5) is at a (-2, 2) b (-4, -8) c (-1, 1) d (1, -1) 2. the length of the line segment with endpoints c(-5, 2) and d(1, -4) is a $sqrt{20}$ b $sqrt{24}$ c $sqrt{72}$ d $sqrt{80}$

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For points $A(-3,-3)$ and $B(1,5)$, $x_1=-3,y_1 = - 3,x_2=1,y_2 = 5$.

Step2: Calculate x - coordinate of mid - point

$x=\frac{-3 + 1}{2}=\frac{-2}{2}=-1$.

Step3: Calculate y - coordinate of mid - point

$y=\frac{-3 + 5}{2}=\frac{2}{2}=1$. So the mid - point of segment $AB$ is $(-1,1)$.

Step4: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $C(-5,2)$ and $D(1,-4)$, $x_1=-5,y_1 = 2,x_2=1,y_2=-4$.

Step5: Calculate $(x_2 - x_1)^2+(y_2 - y_1)^2$

$(x_2 - x_1)^2=(1-(-5))^2=(1 + 5)^2=36$, $(y_2 - y_1)^2=(-4 - 2)^2=(-6)^2 = 36$. Then $(x_2 - x_1)^2+(y_2 - y_1)^2=36+36 = 72$.

Step6: Calculate the distance

$d=\sqrt{72}$.

Answer:

1. C. (-1, 1)
2. C. $\sqrt{72}$