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.
+658 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 
+658 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
