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 Length of Angle Bisector (Posted on 2010-11-07)
In the below diagram, various segment lengths in rectangle ABCD have been marked and segment EG bisects angle DEF. What is the length of segment EG?

 See The Solution Submitted by Charlie Rating: 2.0000 (2 votes)

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 Solution | Comment 1 of 3
`|ED| = 17 = |EF|  and`
`angle DEG = angle FEG. Therefore,`
`Triangles DEG and FEG are congruent.`
`Thus, |DG| = |FG|. Therefore,`
`  20^2 + |AG|^2 = |DA|^2 + |AG|^2`
`                = |DG|^2 = |FG|^2`
`                = |GB|^2 + |BF|^2`
`                = |GB|^2 + 5^2`
`                or`
`  |GB|^2 - |AG|^2 = 20^2 - 5^2`
`                or`
`  (|GB| - |AG|)(|GB| + |AG|)`
`  = (|GB| - |AG|)(25) =  (20 - 5)(25)`
`Thus,`
`  |GB| - |AG| = 15`
`  |GB| + |AG| = 25`
`Therefore, |AG| = 5. Thus,`
`  |EG| = sqrt(|DA|^2 + [|DE| - |AG|]^2)`
`       = sqrt(20^2 + 12^2) `
`       = sqrt(544) ~= 23.3238`
` `

 Posted by Bractals on 2010-11-07 14:15:50

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