My Math Forum  

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

Geometry Geometry Math Forum


Thanks Tree1Thanks
  • 1 Post By mijjim
Reply
 
LinkBack Thread Tools Display Modes
July 9th, 2014, 09:13 PM   #1
Newbie
 
Joined: Jul 2014
From: Mars

Posts: 1
Thanks: 0

Right Triangles Proof

Quadrilateral WXYZ has right angles at angle W and angle Y and an acute angle at angle X. Altitudes are dropped from X and Z to diagonal WY, meeting WY at O and P. Prove that OW = PY.
problemsolver432 is offline  
 
July 10th, 2014, 07:51 PM   #2
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 12,419
Thanks: 831

Changing your lettering to (hate using O and Z!):
Quadrilateral ABCD has right angles at A and C and an acute angle at angle B.
Altitudes are dropped from B and D to diagonal AC, meeting AC at E and F.
Prove that AE = CF.

That's same as proving that AF = CE (since EF is common to both AE and CF).

a = AB, b = BC, c = CD, d = DA.
e = CE, f = AF
g = BD, h = AC (diagonals)
m = BE, n = DF (heights)

Prove that e = f.

triangleABD: g^2 = a^2 + d^2
triangleBCD: g^2 = b^2 + c^2
a^2 + d^2 = b^2 + c^2
a^2 + d^2 - b^2 - c^2 = 0 [1]

triangleBCE: m^2 = b^2 - e^2
triangleABE: m^2 = a^2 - (h - e)^2
b^2 - e^2 = a^2 - (h - e)^2
Simplifies to:
h^2 = a^2 - b^2 + 2eh [2]

triangleADF: n^2 = d^2 - f^2
triangleCDF: n^2 = c^2 - (h - f)^2
d^2 - f^2 = c^2 - (h - f)^2
Simplifies to:
h^2 = c^2 - d^2 + 2fh [3]

[2][3]:
a^2 - b^2 + 2eh = c^2 - d^2 + 2fh
a^2 + d^2 - b^2 - c^2 = 2fh - 2eh
[1]: 0 = 2h(f - e)
Since h <>0, then f - e = 0, so e = f

Last edited by Denis; July 10th, 2014 at 07:56 PM.
Denis is offline  
October 24th, 2014, 11:57 PM   #3
Newbie
 
Joined: Oct 2014
From: Lipowo

Posts: 1
Thanks: 1

Let be a point in the intersection of the line and the circumcircle of . Inscribed angles and are equal so . Therefore (or ) is isosceles trapezoid.
Thanks from aurel5
mijjim is offline  
Reply

  My Math Forum > High School Math Forum > Geometry

Tags
geometry, homework, proof, triangles



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Triangles Dacu Algebra 0 May 25th, 2013 11:28 PM
Triangles Archonn Algebra 20 November 5th, 2012 12:35 PM
30-60-90 Triangles? Recipe Algebra 9 March 13th, 2010 08:19 AM
sin,cos,tan and triangles Malgrif Algebra 2 May 29th, 2008 04:36 AM
triangles tejolson Computer Science 1 December 31st, 1969 04:00 PM





Copyright © 2018 My Math Forum. All rights reserved.