Maximizing Intercepted Lengths in a Right Triangle Inscribed in a Circle

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SUMMARY

The discussion focuses on maximizing the length intercepted (A+B) by a right triangle inscribed in a circle with radius 'a'. The key insight involves using the Pythagorean theorem and relating the triangle's dimensions to the circle's properties. The diameter of the circle is established as 2a, and the angle θ is introduced as a crucial variable for deriving the relationship between A and B. The solution requires a geometric approach to optimize the lengths A and B effectively.

PREREQUISITES
  • Understanding of the Pythagorean theorem
  • Basic knowledge of circle geometry
  • Familiarity with trigonometric functions
  • Ability to manipulate variables in geometric contexts
NEXT STEPS
  • Explore the relationship between angles and intercepted lengths in circle geometry
  • Learn about optimization techniques in geometric problems
  • Study the properties of inscribed angles and triangles
  • Investigate the application of trigonometric identities in maximizing functions
USEFUL FOR

Students studying geometry, mathematics educators, and anyone interested in solving optimization problems involving circles and triangles.

ptolema
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Homework Statement



A right angle is moved along the diameter of a circle of radius a (see diagram). What is the greatest possible length (A+B) intercepted on it by the circle.

fig27.jpg


Homework Equations



so, the pythagorean theorem might be useful
diameter = 2a

The Attempt at a Solution



i have to maximise A+B, but i don't exactly have an equation to do that. maybe maximising A2+B2 would work, but that still leaves me with too many variables. i don't know how to relate anything from the circle to the right angle besides the obvious diameter. i know that 0<A<a and 0<B<2a, but this once again gets me nowhere. no idea where to start, please help!
 
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hi ptolema! :smile:

hint: draw the line from A to the centre of the circle, and call the angle there θ. :wink:
 
thanks, that was a big help!
 

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