My Math Forum  

Go Back   My Math Forum > High School Math Forum > Geometry

Geometry Geometry Math Forum

Thanks Tree3Thanks
  • 3 Post By mrtwhs
LinkBack Thread Tools Display Modes
April 22nd, 2018, 07:25 AM   #1
Joined: Jan 2018
From: Belgrade

Posts: 55
Thanks: 2

Minimal value of the sum of two line segments

$\displaystyle ABCD$ is square with side $\displaystyle a$. There is a point E on side $\displaystyle AB$, so that $\displaystyle AE : EB = 2 : 1$. Let F be an arbitrary point on diagonal $\displaystyle BD$. Prove that $\displaystyle AF + EF \geqslant \frac{a \sqrt{10}}{3}$.
I proved that $\displaystyle AF + EF \geqslant \frac{a \sqrt{12}}{3}$.
It's the case when $\displaystyle AF \geqslant \frac{AC}{2}$, and $\displaystyle EF \geqslant \frac{a}{3\sqrt{2}}$ (when $\displaystyle EF \perp BD$).
Do you know how to prove the original inequality?
Attached Images
File Type: png Drz2016_VII_z5.png (6.0 KB, 5 views)

Last edited by lua; April 22nd, 2018 at 07:27 AM.
lua is offline  
April 24th, 2018, 03:13 AM   #2
Senior Member
mrtwhs's Avatar
Joined: Feb 2010

Posts: 711
Thanks: 147

Reflect point $\displaystyle E$ around diagonal $\displaystyle BD$ to hit side $\displaystyle BC$ at point $\displaystyle G$. Connect $\displaystyle AG$ hitting the diagonal at point $\displaystyle F$.

Now $\displaystyle AF+EF=AF+FG=AG$. By the Pythagorean Theorem $\displaystyle AG = \dfrac{a}{3}\sqrt{10}$.

Sorry my picture is upside down from yours.
Attached Images
File Type: jpg picture.jpg (8.2 KB, 0 views)
Thanks from johng40, JeffM1 and lua

Last edited by mrtwhs; April 24th, 2018 at 03:21 AM.
mrtwhs is offline  
April 24th, 2018, 04:51 AM   #3
Joined: Jan 2018
From: Belgrade

Posts: 55
Thanks: 2

O, great. I see. If $\displaystyle F$ was anywhere else on $\displaystyle BD$, it would be:
$\displaystyle AF+EF=AF+FG>AG$.
So, $\displaystyle AG$ is minimal value of $\displaystyle AF+EF$.

Thank you, very much.
lua is offline  

  My Math Forum > High School Math Forum > Geometry

line, minimal, segments, sum

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Calculating the line integral of a curve given line segments nomad609 Calculus 0 October 23rd, 2016 11:39 AM
Making a Circle out of line segments Jmlee19 Calculus 5 December 6th, 2012 02:23 PM
Geometry piecewise line segments master555 Applied Math 0 December 2nd, 2011 05:56 PM
Line segments-distance andoxx Algebra 7 February 8th, 2011 08:09 PM
Overlap of line segments Cheesy74 Algebra 4 November 17th, 2009 05:23 AM

Copyright © 2019 My Math Forum. All rights reserved.