Circle A has a radius of #3 # and a center of #(2 ,5 )#. Circle B has a radius of #3 # and a center of #(3 ,8 )#. If circle B is translated by ##, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Question

Circle A has a radius of #3 # and a center of #(2 ,5 )#.  Circle B has a radius of #3 # and a center of #(3 ,8 )#.  If circle B is translated by #<4 ,2 >#, does it overlap circle A?  If not, what is the minimum distance between points on both circles?

Aaliyah Anderson · Accepted Answer

no overlap , ≈ 1.07

What we have to do here is compare the distance (d) between the centres with the sum of the radii. • If sum of radii > d , then circles overlap • If sum of radii < d , then no overlap First we require to find the new position of centre B. A translation does not change the shape of a figure only it's position. Under a translation of #((4),(2))# centre of B(3 ,8) → (3+4 ,8+2) → (7 ,10) To calculate d , use the #color(blue)" distance formula "# #color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|))# where # (x_1,y_1)" and " (x_2,y_2)" are 2 points"# let # (x_1,y_1)=(2,5)" and " (x_2,y_2)=(7,10)# #d=sqrt((7-2)^2+(10-5)^2)=sqrt50 ≈ 7.07# radius of A + radius of B = 3 + 3 = 6 Since sum of radii < d , then no overlap and minimum distance = 7.06 - 6 = 1.06 graph{(y^2-10y+x^2-4x+20)(y^2-20y+x^2-14x+140)=0 [-35.56, 35.56, -17.78, 17.78]}

Ella Armstrong · Answer

undefined The translated center of Circle B is (3 + 4, 8 + 2) = (7, 10). To determine if Circle B overlaps Circle A, we calculate the distance between their centers and compare it to the sum of their radii. The distance between the centers is $\sqrt{(7 - 2)^2 + (10 - 5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5$. The sum of their radii is 3 + 3 = 6. Since the distance between the centers (5) is less than the sum of their radii (6), the circles do overlap. If you meant to ask for the minimum distance between points on both circles, it would be the difference between the distance of their centers (5) and the sum of their radii (6), which is 1 unit.

Circle A has a radius of #3 # and a center of #(2 ,5 )#. Circle B has a radius of #3 # and a center of #(3 ,8 )#. If circle B is translated by #&lt;4 ,2 &gt;#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Circle A has a radius of #3 # and a center of #(2 ,5 )#. Circle B has a radius of #3 # and a center of #(3 ,8 )#. If circle B is translated by #<4 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?