A geometric proof (minimizing the length of two lines in a rectangle)

In summary, The conversation discusses using calculus and geometry to prove a specific equation. It also mentions flipping a line to create a rectangle and finding the shortest line between two points. A suggestion is also requested for further clarification.
  • #1
ali PMPAINT
44
8
Homework Statement
Show that in a rectrangular ABCD, with a point P on CD, Show that AP+BP is smallest when CP=DP
Relevant Equations
Maybe sin(a+b)=sin(a)cos(b)+cos(a)sin(b) and sin(a-b)=sin(a)cos(b)-cos(a)sin(b)
So, I know it can be proven using calculus, but I need the geometric one.
244525

So, I got that ^c=^d and therefor, the amount of increment in one of a, is equal to the other(^e=^b). (Also 0<a+b<Pi/2)
And AP'=BP'=BD/sin(a) and BP=BD/sin(a+b) and AP=BD/sin(a-b).
AP'+BP'=2AP'=2BD/sin(a) and AP+BP=BD(1/sin(a+b)+1/sin(a-b))
Know I'm trying to show 1/sin(a+b)+1/sin(a-b)>2/sin(a) for 0<a+b<Pi/2 , But no result.
Any suggestion please?
 

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  • #2
Flip over the line CD to get A' and B' as corners of rectangle A'B'B A
What's the shortest line from A'to B ?
 
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  • #3
BvU said:
Flip over the line CD to get A' and B' as corners of rectangle A'B'B A
What's the shortest line from A'to B ?
Thanks for the hint!
244544

On triangle BP'A', we have BP'+A'P'>BA' and beacuase BA'=2BO=AO+BO , and BP'+A'P'=BP'+AP' , we get BP'+AP'>AO+BO
 
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Related to A geometric proof (minimizing the length of two lines in a rectangle)

1. What is a geometric proof?

A geometric proof is a logical argument that uses geometric principles and properties to demonstrate the truth of a statement or theorem.

2. What is meant by "minimizing the length of two lines in a rectangle"?

This refers to finding the shortest possible lengths for two lines within a rectangle while still maintaining the same area. It is typically used to optimize the dimensions of a rectangle for a given area.

3. What are the steps involved in a geometric proof for minimizing the length of two lines in a rectangle?

The steps involved in a geometric proof for minimizing the length of two lines in a rectangle typically include identifying the given information, setting up equations or inequalities, applying relevant geometric principles and theorems, and arriving at a logical conclusion.

4. What are some real-world applications of minimizing the length of two lines in a rectangle?

Some real-world applications of this concept include optimizing the dimensions of a garden or field, minimizing the amount of fencing needed for a given area, and maximizing the space in a room by minimizing the length of two walls.

5. How does minimizing the length of two lines in a rectangle relate to other geometric concepts?

This concept is related to other geometric concepts such as optimization, area and perimeter, and the Pythagorean theorem. It also involves using properties of rectangles and other shapes to find the most efficient solution.

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