SOLUTION: two circles intersect (a nd b) and share a common chord (cd). the radius of circle a is 10in, radius circle b = 16 in. distance between centers, 22 in. find cd.

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Question 1207163: two circles intersect (a nd b) and share a common chord (cd). the radius of circle a is 10in, radius circle b = 16 in. distance between centers, 22 in. find cd.
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
circle a:
r=10in
circle b:
r=16in
distance between centers=22in
if we place the center of the circle a on x-ais, at point (-11,0), than the center of the circle b will be at (11,0) and the distance between them is 22in

now we have equations:
circle a
%28x-%28-11%29%29%5E2%2B%28y-0%29%5E2=10%5E2+
%28x%2B11%29%5E2%2By%5E2=100

circle+b
%28x-11%29%5E2%2B%28y-0%29%5E2=16%5E2+
%28x-11%29%5E2%2By%5E2=256

find intersection points by solving this system:
%28x-11%29%5E2%2By%5E2=256
%28x%2B11%29%5E2%2By%5E2=100
-----------------------------subtract
%28x-11%29%5E2%2By%5E2-%28x%2B11%29%5E2-y%5E2=256-100
%28x-11%29%5E2-%28x%2B11%29%5E2=156
x%5E2+-+22x+%2B+121-%28x%5E2+%2B+22x+%2B+121%29=156
x%5E2+-+22x+%2B+121-x%5E2+-22x+-+121=156
-+22x++-22x+=156
-+44x+=156
x=156%2F-44
x=-39%2F11


plug it in
%28-39%2F11%2B11%29%5E2%2By%5E2=100
6724%2F121%2By%5E2=100
y%5E2=100-6724%2F121
y%5E2=5376%2F121
y=sqrt%285376%2F121%29
y+=+%2816+sqrt%2821%29%29%2F11+or y+=-+%2816+sqrt%2821%29%29%2F11

solutions:
x+=+-39%2F11, y+=+%2816sqrt%2821%29%29%2F11
or
x+=+-39%2F11, y+=+-%2816sqrt%2821%29%29%2F11

intersection points are
(-39%2F11, %2816sqrt%2821%29%29%2F11) and (-39%2F11, -%2816sqrt%2821%29%29%2F11)

distance between is: two times y value
d=2%2A%2816sqrt%2821%29%29%2F11=13.33

The length of cd is 13.33in.



Answer by ikleyn(52925) About Me  (Show Source):
You can put this solution on YOUR website!
.
two circles intersect (A nd B) and share a common chord (CD).
the radius of circle a is 10in, radius circle b = 16 in. distance between centers, 22 in.
find cd.
~~~~~~~~~~~~~~~~~~~ 

Draw the radii of the circles from their centers to the intersection point C.
Connect the centers by the straight line AB.


You will get a triangle ABC with the sides 10 in, 16 in and 22 in.


Find its area using the Heron's formula

    area = sqrt%28s%2A%28s-10%29%2A%28s-16%29%2A%28s-22%29%29,

where s = %2810%2B16%2B22%29%2F2%29 = 24 is the semi-perimeter.


Substituting this number into the formula, you will get

    area = sqrt%2824%2A%2824-10%29%2A%2824-16%29%2A%2824-22%29%29 = sqrt%2824%2A14%2A8%2A2%29 = sqrt%283%2A8%5E2%2A7%2A4%29 = 

         = 8%2A2%2Asqrt%2821%29 = 16%2Asqrt%2821%29 = 73.32121112... square inches.


Write the formula for the area of triangle ABC in other way, using the base AB = 22 in and the height h
from point C to the base

    area = %281%2F2%29%2A22%2Ah = 11*h  square inches.


You will get an equation

    11*h = 16%2Asqrt%2821%29.

Hence,  

    h = %2816%2F11%29%2Asqrt%2821%29%29.


This value of h is half of the length CD;  so

    CD = %2832%2F11%29%2Asqrt%2821%29 = 13.331 inches  (rounded).


At this point, the problem is just solved.


ANSWER.  The length CD is  %2832%2F11%29%2Asqrt%2821%29 = 13.331 inches  (rounded).

Solved.