Circle A has a center at #(2 ,2 )# and an area of #18 pi#. Circle B has a center at #(13 ,6 )# and an area of #27 pi#. Do the circles overlap?

Question

Circle A has a center at #(2   ,2  )# and an area of #18 pi#.  Circle B has a center at #(13  ,6 )# and an area of #27 pi#.  Do the circles overlap?

Allison Allen · Accepted Answer

There is no Overlap

so we have two circles,

A, with centre #(2,2)# and Area #18pi#
B, with centre #(13,6)# and Area #27pi#

We will work with A first

#A=pir^2#

#18pi=pir^2#

#r=sqrt(18) = 3*sqrt(2)#

Now B,

#27pi=pir^2#

#r=sqrt(27) = 3*sqrt(3)#

so if they overlap the distance between the centres of the circles will be less than the two radii.

Distance between the two circles,

#vec(AB)#=#B-A#

#=(13,6)-(2,2)#

#=(11,4)#

Using Pythagoras theorem,

Distance = #sqrt(a^2+b^2)#

#=sqrt(11^2+4^2)#

#=sqrt(137)#

Now to find out,
To overlap this must be true,

#sqrt(137) < r_"A"+r_"B"#

#sqrt(137) < 3*sqrt(2)+3*sqrt(3)#

#11.7047<4.2426+5.1962#

#11.7047<9.4388#

This is not true so there is no Overlap.

Visually,

The line f is longer than the two radii combined.

Theodore Abad · Answer

undefined To determine if the circles overlap, we can compare the distances between their centers to the sum of their radii.

Step 1: Find the radii of each circle using the given areas.
Step 2: Calculate the distance between the centers of the circles using the distance formula.
Step 3: Check if the distance between the centers is less than the sum of the radii. If it is, then the circles overlap; otherwise, they do not overlap.

After calculating the radii of the circles and the distance between their centers, compare these values to determine if the circles overlap.