Question

1. a ranger tracks a bear from its den at (-8,-4) to a food source at (6, 10). a) how far did the bear travel? b) where is the midpoint, representing the halfway point of its journey?

Explanation:

Step1: Recall distance formula

The distance $d$ 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, $(x_1,y_1)=(-8,-4)$ and $(x_2,y_2)=(6,10)$.

Step2: Substitute values into distance formula

$d=\sqrt{(6 - (-8))^2+(10 - (-4))^2}=\sqrt{(6 + 8)^2+(10 + 4)^2}=\sqrt{14^2+14^2}=\sqrt{196 + 196}=\sqrt{392}=14\sqrt{2}\approx19.8$

Step3: Recall mid - point formula

The mid - point $M$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step4: Substitute values into mid - point formula

$M=(\frac{-8+6}{2},\frac{-4 + 10}{2})=(\frac{-2}{2},\frac{6}{2})=(-1,3)$

Answer:

a) The bear traveled $14\sqrt{2}\approx19.8$ units.
b) The mid - point is $(-1,3)$.