Q $7$ $\newline$ $D(4.5,3), E(5,11)$ and $F(2.5,5)$ are three points in the $x y$ -plane. If $\overline{D F}$ is a diameter of Circle $1$ and $\overline{E F}$ is a diameter of Circle $2$ , what is the slope of the line that goes through the centers of the two circles?

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Q. Q

7

\newline

D(4.5,3), E(5,11)

and

F(2.5,5)

are three points in the

x y

-plane. If

\overline{D F}

is a diameter of Circle

1

and

\overline{E F}

is a diameter of Circle

2

, what is the slope of the line that goes through the centers of the two circles?

Find Midpoints of Diameters: To find the centers of the circles, we need to find the midpoints of the diameters $DF$ and $EF$ .
Calculate Midpoint M $1$ : The midpoint $M_1$ of $DF$ can be found using the midpoint formula: $M_1 = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ , where $D(4.5,3)$ and $F(2.5,5)$ .
Calculate Midpoint M $2$ : Calculating the midpoint $M1$ : $M1 = ((4.5 + 2.5)/2, (3 + 5)/2) = (7/2, 8/2) = (3.5, 4)$ .
Find Centers of Circles: The midpoint $M_2$ of $EF$ can be found using the midpoint formula: $M_2 = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ , where $E(5,11)$ and $F(2.5,5)$ .
Calculate Slope: Calculating the midpoint $M_2$ : $M_2 = ((5 + 2.5)/2, (11 + 5)/2) = (7.5/2, 16/2) = (3.75, 8)$ .
Calculate Slope: Calculating the midpoint $M_2$ : $M_2 = ((5 + 2.5)/2, (11 + 5)/2) = (7.5/2, 16/2) = (3.75, 8)$ .Now we have the coordinates of the centers of both circles: $M_1(3.5, 4)$ and $M_2(3.75, 8)$ .
Calculate Slope: Calculating the midpoint $M_2$ : $M_2 = ((5 + 2.5)/2, (11 + 5)/2) = (7.5/2, 16/2) = (3.75, 8)$ .Now we have the coordinates of the centers of both circles: $M_1(3.5, 4)$ and $M_2(3.75, 8)$ .The slope of the line through the centers of the two circles can be found using the slope formula: $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$ , where $M_1(3.5, 4)$ and $M_2(3.75, 8)$ .
Calculate Slope: Calculating the midpoint $M_2$ : $M_2 = ((5 + 2.5)/2, (11 + 5)/2) = (7.5/2, 16/2) = (3.75, 8)$ .Now we have the coordinates of the centers of both circles: $M_1(3.5, 4)$ and $M_2(3.75, 8)$ .The slope of the line through the centers of the two circles can be found using the slope formula: $\text{slope} = (y_2 - y_1) / (x_2 - x_1)$ , where $M_1(3.5, 4)$ and $M_2(3.75, 8)$ .Calculating the slope: $\text{slope} = (8 - 4) / (3.75 - 3.5) = 4 / 0.25 = 16$ .