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Expert: - 6/10/2010


Question

Geometry Problem
Hello sir, and good evening.  My name is Charles, and I am an 8th grade student taking Honors Geometry.  I need help with one problem that I've been trying to work out all week, but an answer or a way to solve the problem has simply been elusive to me.  I was hoping that you could help me.  
Best Regards,
Charles McCullough

PS:  Thanks in advance!  The problem is the attached image, I drew it then scanned it into my computer!

Answer
When points A and D have been picked and are on the diameter of a circle, B is on the circle since the angle at B is a right angle (http://en.wikipedia.org/wiki/Right_triangle).  This means that on the line that is 16 long, the center of the circle can be found 6 untis from the left (or 10 from the right) since 16+4=20, which is the circles diameter.  The radius, then, is 20/2=10, and that's where the center is, which we'll call E.

This means another triangle can be drawn using B, C, and E.  It is known that EB is the hypoteneuse and CE is one leg, so x is the other leg.  It can be seen that EB has length 10.
It can also be seen that CE has length 6.  Since BCE is a right triangle, it is known that
x² + 6² = 10².

Form here, I'm sure that you can find the value for x.

Hint: this is a special case where x happens to also the average of 6 and 10...
but that's only in triangles like this.

Scott A Wilson

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