Question

#1.) mr. longo is trying to find the most convenient place to park his car so that its equidistant from his gym and his house. his house is located at h(-4,7) and his gym is at g(6,1). graph \\(\overline{hg}\\) and use it to answer the questions below.
a.) **determine the coordinate point c where mr. longo should park his car so its an even distance from the gym to his house. plot it on the graph and state it. the use of the coordinate plane below is optional.
b.) *how far is mr. longos walk from his gym to where he should park his car? think: how do we find distance between two points?
#2.) explain why (0,0) cannot be the midpoint of (-3,4) and (3,4). use the grid to support your evidence.

Explanation:

Step1: Find the mid - point for part a

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Given $H(-4,7)$ and $G(6,1)$, we have $x_1=-4,y_1 = 7,x_2=6,y_2 = 1$.
$x=\frac{-4 + 6}{2}=\frac{2}{2}=1$
$y=\frac{7+1}{2}=\frac{8}{2}=4$
So the mid - point $C$ is $(1,4)$.

Step2: Find the distance for part b

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 $G(6,1)$ and $C(1,4)$, we have $x_1 = 6,y_1=1,x_2 = 1,y_2 = 4$.
$d=\sqrt{(1 - 6)^2+(4 - 1)^2}=\sqrt{(-5)^2+3^2}=\sqrt{25 + 9}=\sqrt{34}$

Step3: Explain for part 2

Using the mid - point formula for points $(-3,4)$ and $(3,4)$, $x=\frac{-3+3}{2}=0$ and $y=\frac{4 + 4}{2}=4$. The mid - point is $(0,4)$ not $(0,0)$. On the grid, if we plot $(-3,4)$ and $(3,4)$, we can see that the vertical line of symmetry is at $x = 0$ and the mid - point has a $y$ - coordinate of 4, not 0.

Answer:

a. $(1,4)$
b. $\sqrt{34}$

1. The mid - point of $(-3,4)$ and $(3,4)$ is $(0,4)$ not $(0,0)$ as calculated by the mid - point formula and observed on the grid.