All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Bicentric Trapezoids (Posted on 2010-07-24)
A certain trapezoid happens to be a bicentric quadrilateral. The trapezoid's circumradius is twice the inradius. What are the measures of the angles of the trapezoid?

If the circumradius is five times the inradius, then what are the measures of the angles?

What is the ratio of the circumradius and inradius when the longer base of the trapezoid is a diameter of the circumcircle?

 See The Solution Submitted by Brian Smith Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution Comment 1 of 1
`Let r, R, and s be the lengths of theinradius, circumradius, and the distancebetweeen the incenter and circumcenterrespectively. The relation between thesethree is`
`   (R^2 - s^2)^2 = 2*r^2*(R^2 + s^2).  (*)`
`Solving for s^2 gives`
`   s^2 = [r^2 + R^2] - r*sqrt{r^2 + 4*R^2]     `
`Let the shorter of the two parallel lineshave length 2a and the longer 2b. Clearly,the slant sides have an equal length ofa + b.`
`Using the Pathagorean theorem we have`
`   a^2 = R^2 - (r + s)^2`
`   b^2 = R^2 - (r - s)^2`
`   r^2 = a*b`
`The four angles of the trapezoid have two values that are supplementary. Thesmaller of these two values is`
`   arccos([b - a]/[b + a])      = arccos([b^2 - a^2]/[b + a]^2)     = arccos(2*r*s/[R^2 - s^2])`
`With R = kr,`
`   s^2 = r^2*[(k^2 + 1) - sqrt(4*k^2 + 1)]`
`   2*r*s = 2*r^2*(sqrt[4*k^2 + 1] + 1)*           sqrt[(k^2 + 1) - sqrt(4*k^2 + 1)]`
`   R^2 - s^2 = r^2*[sqrt(4*k^2 + 1) - 1]`
`   arccos(2*r*s/[R^2 - s^2]) =`
`     arccos(sqrt[([2*k^2 - 1] -       sqrt[4*k^2 + 1])/(2*k^2)]`
`Problem#1 (k = 2)`
`   arccos(sqrt[(7 - sqrt[17])/8]) `
`          ~= 53.1533 deg.`
`Problem#2 (k = 5)`
`   arccos(sqrt[(49 - sqrt[101])/50]) `
`          ~= 28.0410 deg.`
`Problem#3 (2*R = 2*b  <==>  s = r)`
`   Solving equation (*) for R,`
`     (R^2 - r^2)^2 = 2*r^2*(R^2 + r^2)`
`                   or`
`     R^2 = r^2*(2 + sqrt[5])`
`   Therefore,`
`     R/r = sqrt(2 + sqrt[5]) ~= 2.0582`
` `

 Posted by Bractals on 2010-07-24 12:45:46

 Search: Search body:
Forums (0)