Two parallel chords in a circle have lengths 6 and 8. The distance between them is 1. Find the diameter of the circle.
Two parallel chords in a circle have lengths 6 and 8.
The distance between them is 1.
Find the diameter of the circle.
\(\begin{array}{|rcll|} \hline 3^2+(1+x)^2 &=& R^2 \\ 4^2+x^2&=& R^2 \\ \hline R^2 = 3^2+(1+x)^2 &=& 4^2+x^2 \\ 3^2+(1+x)^2 &=& 4^2+x^2 \\ 9+1+2x+x^2 &=& 16+x^2 \\ 9+1+2x &=& 16 \\ 10+2x &=& 16 \\ 2x &=& 6 \\ \mathbf{x} &=& \mathbf{3} \\\\ R^2 &=& 4^2 + x^2 \\ R^2 &=& 16+3^2 \\ R^2 &=& 25 \\ \mathbf{R} &=& \mathbf{5} \\ \mathbf{2R} &=& \mathbf{10} \\ \hline \end{array}\)
The diameter of the circle is 10
If we bisect both chords from the center of the circle and draw a radius to both chord ends, we will have two right triangles.....
Call the distance from the center of the circle to the longer chord = d
Call the distance from the center of the circle to the shorter chord = d + 1
So....we have this system
4^2 + d^2 = r^2
3^2 + (d + 1)^2 = r^2
Set the r's equal
4^2 + d^2 = 3^2 + (d + 1)^2
16 + d^2 = 9 + d^2 + 2d + 1
16 = 10 + 2d
6 = 2d
d = 3
And using either equation we have that
4^2 + d^2 =r^2
16 + 9 = r^2
25 = r^2
r = 5
So....the diameter = 10