My Math Forum  

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

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
October 29th, 2013, 12:50 PM   #1
Newbie
 
Joined: Sep 2013

Posts: 10
Thanks: 0

Pythagoras problem

I tried using Apollonius's theorem to relate v, w and CP.
Let the side opposite <B be b and <A be a and v+w be c. Let D be the point on AB where the altitude from <C falls and PD=y.
By Pythagoras:
b^2+a^2=v^2+w^2+2CP^2
=(v-y)^2+(v+y)^2+2CP^2
=2(v+y)^2+2y^2+2CP^2
this is where I get stuck I see that b^2+a^2=(v+w)^2 but I can't simplify it to get that v^2+w^2=2CP^2
Attached Images
File Type: jpg image.jpg (49.7 KB, 169 views)
Raye is offline  
 
October 29th, 2013, 06:19 PM   #2
Senior Member
 
Joined: Jul 2011

Posts: 118
Thanks: 0

Re: Pythagoras problem

octahedron is offline  
October 29th, 2013, 06:42 PM   #3
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,949
Thanks: 1141

Math Focus: Elementary mathematics and beyond
Re: Pythagoras problem





From the law of cosines,





Similarly,



Adding these results,

greg1313 is offline  
October 29th, 2013, 09:01 PM   #4
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,949
Thanks: 1141

Math Focus: Elementary mathematics and beyond
Re: Pythagoras problem

Construct a segment from C to AB, bisecting AB at D.

CP = (AD - v) + (w - PD)
= AD - 2vAD + v + w - 2wPD + PD
= v + w + CP - 2(vAD + wPD)
= v + w + CP - 2(v + 2vPD + 2PD) [as AD = PD + v and w = 2PD + v]
= v + w + CP - 2((v + PD) + PD)
= v + w + CP - 2CP
2CP = v + w.
greg1313 is offline  
October 29th, 2013, 09:15 PM   #5
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 14,597
Thanks: 1038

Re: Pythagoras problem

Skinning the cat #3!
Code:
                      C




A           P         D                   B
AC = BC = a (hate "s"!), AP = v, PD = y, BD = w - y, CP = p, CD = d

2a^2 = v^2 + 2vw + w^2 [1]

v + y = w - y : y = (w - v) / 2 [2]

d^2 = p^2 - y^2 [3]

d^2 = a^2 - (w - y)^2 : d^2 = a^2 - w^2 + 2wy - y^2 [4]

[3][4]: p^2 = a^2 - w^2 + 2wy
Substitute [2]:
p^2 = a^2 - w^2 + 2w((w - v)/2) : 2p^2 = 2a^2 - 2vw
Substitute [1]:
2p^2 = v^2 + 2vw + w^2 - 2vw : 2p^2 = v^2 + w^2
Denis is offline  
October 30th, 2013, 02:13 AM   #6
Global Moderator
 
Joined: Dec 2006

Posts: 20,804
Thanks: 2149

Construct PD perpendicular to BC with D on BC.
It is easy to show that PD = w/?2 and CD = v/?2.
By Pythagoras, v/2 + w/2 = CP, so v + w = 2CP.
skipjack is online now  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
problem, pythagoras



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
PYTHAGORAS COORDINATES stevembe Algebra 10 September 23rd, 2013 08:25 AM
Please help me with this Pythagoras Theorem Question nadiva8685 Algebra 3 January 22nd, 2013 04:37 PM
pythagoras triplet kaushiks.nitt Number Theory 2 June 27th, 2009 06:51 PM
How did Pythagoras theorem come about? graviton120 Algebra 3 October 31st, 2008 08:04 PM
Two questions regarding Sin/Cos & Pythagoras jsmars Algebra 1 May 26th, 2008 01:53 AM





Copyright © 2019 My Math Forum. All rights reserved.