Maximizing Intercepted Lengths in a Right Triangle Inscribed in a Circle

In summary, the question asks for the maximum possible length (A+B) intercepted on a right angle moved along the diameter of a circle of radius a. The Pythagorean theorem may be useful, and the diameter is equal to 2a. Drawing a line from A to the center of the circle and labeling the angle there as θ can help solve the problem.
  • #1
ptolema
83
0

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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  • #2
hi ptolema! :smile:

hint: draw the line from A to the centre of the circle, and call the angle there θ. :wink:
 
  • #3
thanks, that was a big help!
 

Related to Maximizing Intercepted Lengths in a Right Triangle Inscribed in a Circle

1. What is the Triangle Optimization problem?

The Triangle Optimization problem is a mathematical problem that involves finding the optimal solution for a triangle with given constraints, such as side lengths, angles, or area.

2. What are the applications of Triangle Optimization?

Triangle Optimization has various applications in fields such as engineering, architecture, and computer graphics. It can be used to design efficient and stable structures, create realistic 3D models, and optimize resource usage.

3. How is the Triangle Optimization problem solved?

The Triangle Optimization problem is typically solved using mathematical techniques such as calculus, geometry, and linear algebra. Various algorithms, including the Golden Section search and the Newton's method, can also be used to find the optimal solution.

4. What are the common constraints in the Triangle Optimization problem?

Some common constraints in the Triangle Optimization problem include fixed side lengths, fixed angles, and a fixed area. Other constraints can also be added, such as a maximum or minimum perimeter, to further optimize the triangle.

5. How does Triangle Optimization relate to real-life situations?

Triangle Optimization can be applied to real-life situations, such as optimizing the shape of a sail for a boat, minimizing the material used in a bridge, or finding the most efficient way to cut a piece of fabric. It can also be used in computer graphics to create realistic and visually appealing 3D models.

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