Question 1034430
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Two parallel chords, one 10 and the other 16 and 13 units apart. Find the diameter. 
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Unclear.
Read your post.
How many chords?
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<U>Comment from student</U>: The first thing I said was two chords, there 2 on opposite sides of the circle. 
The one closer to the top is 10 units and the bottom chord is 16. The chords are 13 units apart. 
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My response:

First formulation was BAD. The second makes things clearer, although is not perfect.
But let us turn to the solution.


<pre>
Please make a sketch.  Draw the diameter parallel to the chords.
There are TWO possible configurations. 
1) The chords are in one side of the diameter.
2) The chords are in different sides.

I will consider here only the configuration #1, leaving #2 to you.

Let "x" be the distance from the center to the first chord and "y" be the distance from the center to the first chord.

Then you have this system of 3 equations 

x + y = 13,          (1)     (in configuration #2 it is   x - y = 13)
{{{x^2 + 5^2}}} = {{{r^2}}},      (2)
{{{y^2 + 8^2}}} = {{{r^2}}}.      (3)

Here 5 in (2) is half of 10, while 8 in (3) is half of 16.

Distract equation (3) from (2)  (both sides). You will get

{{{x^2 - y^2)}}} = {{{8^2 - 5^2}}},  or

(x+y)*(x-y) = 64 - 25,  or

(x+y)*(x-y) = 39.     (4)

In (4), replace x+y by 13 due to (1). You will get

x - y = {{{39/13}}} = 3.

Now you have two equations:

x + y = 13.      ( it is former equation (1) )
x - y =  3.

Add them to get 2x = 16 and x = 8.
Then y = 5.

The problem is solved (in half, i.e. for configuration #1).

For #2 please do it yourself following the same scheme.
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