+0  
 
0
169
5
avatar

Chords UV, WX, and YZ of a circle are parallel. The distance between chords UV and WX is 4. The distance between chords WX and YZ is also 4. If UV = 78 and YZ = 50, then find WX.

 Apr 2, 2020

Best Answer 

 #1
avatar+663 
+1

Hello Guest!

It is always nice to draw auxillary lines.

Set Up:

 - Define the center point of the circle as \(O\)

 - Draw \(UO, YO, ZO, XO\) as the radii of the circle.

 - Draw the height of the isosceles triangles formed by the radii and chords.

 - Define the length of the radii as \(x\)

 - Define the height of triangle \(UOV\) as \(y\)

Plan:

We are going to use the pythagorean theorem to set up a system of equations to find WX.

Equation:

\(x^2-y^2=1521\) (Half of chord UV sqaured)

\(x^2-(y+8)^2=625\) (Half of chord YZ squared)

Solving:

x2 - y2 = 1521

x2 - (y2 + 16y + 64) = 625

 

x2 - y2 = 1521

x2 - y2 - 16y - 64 = 625

Elimination:

16y + 64 = 896

16y = 832

y = 52


Plugging back in, we get:

x^2 - (52)^2 = 1521 = {x=-65, x=65}

Finding WX:

We use pythagorean theorem to find half of WX.

652 - (52 + 4)2 = 1089

sqrt(1089) = 33

 

WX = 33 * 2 = \(\boxed{66}\)

 

P.S. Geometry is my favorite math subject cheeky

 Apr 2, 2020
edited by AnExtremelyLongName  Apr 2, 2020
 #1
avatar+663 
+1
Best Answer

Hello Guest!

It is always nice to draw auxillary lines.

Set Up:

 - Define the center point of the circle as \(O\)

 - Draw \(UO, YO, ZO, XO\) as the radii of the circle.

 - Draw the height of the isosceles triangles formed by the radii and chords.

 - Define the length of the radii as \(x\)

 - Define the height of triangle \(UOV\) as \(y\)

Plan:

We are going to use the pythagorean theorem to set up a system of equations to find WX.

Equation:

\(x^2-y^2=1521\) (Half of chord UV sqaured)

\(x^2-(y+8)^2=625\) (Half of chord YZ squared)

Solving:

x2 - y2 = 1521

x2 - (y2 + 16y + 64) = 625

 

x2 - y2 = 1521

x2 - y2 - 16y - 64 = 625

Elimination:

16y + 64 = 896

16y = 832

y = 52


Plugging back in, we get:

x^2 - (52)^2 = 1521 = {x=-65, x=65}

Finding WX:

We use pythagorean theorem to find half of WX.

652 - (52 + 4)2 = 1089

sqrt(1089) = 33

 

WX = 33 * 2 = \(\boxed{66}\)

 

P.S. Geometry is my favorite math subject cheeky

AnExtremelyLongName Apr 2, 2020
edited by AnExtremelyLongName  Apr 2, 2020
 #2
avatar
0

Thank you so much! Have an excellent day! Stay safe!

Guest Apr 2, 2020
 #3
avatar+663 
+1

Your welcome! Stay safe to you as well!

 #4
avatar+112409 
0

Excellent, AELN  !!!!

 

We do not  have  many people on here that  "do"  Geometry....so....you're a welcome addition  !!!

 

 

cool cool cool

CPhill  Apr 2, 2020
 #5
avatar
0

Excellent work!!! I tried to use trigonometry, but it didn't work. smiley

Guest Apr 2, 2020

55 Online Users

avatar
avatar