QuestionFind the center and equation of the circle for points $A(-1,-2)$ , $B(6,-9)$ , $C(13,-2)$ , and its diameter in feet.

Studdy Solution

STEP 1

Assumptions1. The skydivers form a circular formation. . The coordinates of three skydivers A, B, and C are given as $A(-1,-), B(6,-9)$ and $C(13,-)$ .
3. The center of the skydivers is the center of the circle formed by the skydivers.
4. Each unit represents1 foot.

STEP 2

To find the center of the skydivers, we need to find the center of the circle formed by the skydivers. We can do this by using the formula for the circumcenter of a triangle, which is given by the intersection of the perpendicular bisectors of the sides of the triangle.The midpoint $M$ of a line segment with endpoints $(x1, y1)$ and $(x2, y2)$ is given by $M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)$ The slope of a line segment with endpoints $(x1, y1)$ and $(x2, y2)$ is given by $lope = \frac{y2 - y1}{x2 - x1}$ The slope of the perpendicular bisector of a line with slope $m$ is $-1/m$ .
The equation of a line with slope $m$ and passing through a point $(x0, y0)$ is $y - y0 = m(x - x0)$

STEP 3

First, we find the midpoints of the line segments AB and BC.
For AB, let $A(-1,-2)$ and $B(6,-9)$ be the endpoints.
$M_{AB} = \left(\frac{-1 +6}{2}, \frac{-2 -9}{2}\right)$ For BC, let $B(6,-9)$ and $C(13,-2)$ be the endpoints.
$M_{BC} = \left(\frac{6 +13}{2}, \frac{-9 -2}{2}\right)$

STEP 4

Calculate the midpoints.
$M_{AB} = \left(\frac{}{2}, \frac{-11}{2}\right) = (2., -.)$ $M_{BC} = \left(\frac{19}{2}, \frac{-11}{2}\right) = (9., -.)$

STEP 5

Next, we find the slopes of the line segments AB and BC.
For AB, let $A(-1,-2)$ and $B(,-9)$ be the endpoints.
$lope_{AB} = \frac{-9 - (-2)}{ - (-1)}$ For BC, let $B(,-9)$ and $C(13,-2)$ be the endpoints.
$lope_{BC} = \frac{-2 - (-9)}{13 -}$

STEP 6

Calculate the slopes.
$lope_{AB} = \frac{-}{} = -1$ $lope_{BC} = \frac{}{} =1$

STEP 7

The slopes of the perpendicular bisectors are the negative reciprocals of the slopes of the line segments.
$lope_{perpendicular\, AB} = -\frac{1}{lope_{AB}} = -\frac{1}{-1} =1$ $lope_{perpendicular\, BC} = -\frac{1}{lope_{BC}} = -\frac{1}{1} = -1$

STEP 8

Now, we find the equations of the perpendicular bisectors.
The equation of the perpendicular bisector of AB is $y - (-5.5) =1(x -2.5)$ The equation of the perpendicular bisector of BC is $y - (-5.5) = -1(x -.5)$

STEP 9

implify the equations of the perpendicular bisectors.
For AB $y +5.5 = x -2.5$ For BC $y +5.5 = -x +9.5$

STEP 10

The circumcenter of the triangle, which is the center of the circle, is the intersection of the perpendicular bisectors. We can find this by setting the equations of the perpendicular bisectors equal to each other and solving for $x$ .
$x -2.5 = -x +9.5$

STEP 11

olve for $x$ .
$x =$ $x =6$

STEP 12

Substitute $x =6$ into the equation of the perpendicular bisector of AB to find $y$ .
$y +5.5 =6 -2.5$

STEP 13

olve for $y$ .
$y =6 -2.5 -5.5 = -2$

STEP 14

The center of the skydivers is $(6, -2)$ .
Now, we can find the equation of the circle. The general equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$ First, we need to find the radius. The radius is the distance from the center of the circle to any point on the circle. We can use the distance formula $r = \sqrt{(x2 - x)^2 + (y2 - y)^2}$ Let the center of the circle be $(h, k) = (6, -2)$ and one of the points on the circle be $A(-,-2)$ .
$r = \sqrt{(- -6)^2 + (-2 - (-2))^2}$

STEP 15

Calculate the radius.
$r = \sqrt{(-7)^2 +0} = \sqrt{49} =7$

STEP 16

Now, we can find the equation of the circle. Substitute $h =6$ , $k = -2$ , and $r =$ into the general equation of the circle.
$(x -6)^2 + (y +2)^2 =^2$

STEP 17

implify the equation of the circle.
$(x -6)^2 + (y +2)^2 =49$

STEP 18

The diameter of the circle is twice the radius. Since each unit represents foot, the diameter in feet is $iameter =2r =2 \times7$

STEP 19

Calculate the diameter.
$iameter = \times7 =14\, feet$ a. The approximate coordinate of the center of the skydivers is $(6, -)$ . b. The equation of the circular formation formed by the skydivers is $(x -6)^ + (y +)^ =49$ . c. The diameter of the skydiving formations is14 feet.

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