[AB] and [DE] are diameters of the circle

Question

circle has 5 points 

A (5,3)

B(-1,1)

C is the midpoint 

D(1,5)

E has no coordinates

questions:

i) find the co-ordinates of C, the centre of the circle

ii) find the coordinates of E, the end point of diameter [DE]

iii)find the length of the radius of the circle

iv) find the length of the diameter of the circle

 

Answer 1

Hi Rebecca,

 

Let's begin by finding the coordinates of C (center of circle).  We can use the Midpoint Formula using the endpoints of diameter AB: 

    C(x_,y) = ((x₁ + x₂) / 2,(y₁ + y₂) / 2)

    C(x,y) = ((5 + (-1)) / 2,(3 + 1) / 2)

    C(x,y) = (((5 - 1) / 2), (4 / 2))

    C(x,y) = ((4 / 2),2)

    C(x,y) = (2,2)

 

Next, we can find the coordinates of endpoint E of diameter DE by noting the difference (d) between coordinates of center (C) and the coordinates of endpoint D and then adding that difference to the coordinates of center (C) going in the opposite direction from endpoint D:

    C(2,2) - D(1,5) = d(2 - 1,2 - 5) = d(1,-3)

    E(x,y) = C(2,2) + d(1,-3)

    E(x,y) = E((2 + 1),(2 + (-3)))

    E(x,y) = E(3,(2 - 3))

    E(x,y) = E(3,-1) 

 

Next, we can find the length of the radius by using the distance formula between center (C) and either one of the four diameter endpoints (A, B, D, or E).  Let's use ALL four diameter endpoints to see that all the radii are of equal length:

    AC = √((5 - 2)² + (3 - 2)²)

    AC = √(3² + 1²)

    AC = √(9 + 1)

    AC = √10

    BC = √((-1 - 2)² + (1 - 2)²)

    BC = √((-3)² + (-1)²)

    BC = √(9 + 1)

    BC = √10

    DC = √((1 - 2)² + (5 - 2)²)

    DC = √((-1)² + 3²)

    DC = √(1 + 9)

    DC = √10

    EC = √((3 - 2)² + (-1 - 2)²)

    EC = √(1² + (-3)²)

    EC = √(1 + 9)

    EC = √10

 

Since all 4 radii are shown to have the same length value of √10, we are confident that our calculations for center C and diameter endpoint E are correct.

 

Finally, we can also calculate the length of the diameter since it is simply twice the length of the radius:

    radius = √10

    diameter = 2√10

 

We can also now write the equation of our circle knowing the coordinates of center (h,k) and length of radius (r):

    (x - h)² + (y - k)² = r² [general form]

    (x - 2)² + (y - 2)² = 10 (equation of our circle)

 

Click the link (https://dl.dropbox.com/s/r5xa9xpx7dylnzm/Circle_Plot.png?raw=1) to view a plot of our circle equation with center C(2,2) and diameters AB and DE drawn.  Note that points A, B, D, E are the graphing calculator's best approximations of our given points.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.