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

 (Ortho)Centrally Seeking Length (Posted on 2007-11-06)
T is the orthocenter of a triangle PQR and T is located within the triangle. S is the midpoint of the side QR and Angle QPR = 30o.

The line TS is extended to the point U such that S lies between T and U satisfying TS = SU.

Determine the length of PU given that QR = 1

 See The Solution Submitted by K Sengupta Rating: 2.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution | Comment 1 of 3
`For fixed points Q and R and a fixed angleQPR of 30 degrees, all triangles PQR (withvertex P on the same side of line QR) havethe same circumcircle. Let O be the centerof this circumcircle.`
`Let <XY> denote the vector from point X topoint Y.`
`  <UP> = <OP> - <OU>       = <OP> - (<OT> + <TU>)       = <OP> - <OT> - 2<TS>       = <OP> - <OT> - 2(<OS> - <OT>)       = <OP> + <OT> - 2<OS>       = <OP> + (<OP> + <OQ> + <OR>)         - 2(<OQ> + <OR>)/2       = 2<OP>`
`The restriction that the orthocenter liewithin the triangle PQR is not needed.`
`If we take PQR to be a right triangle, it isclear that the diameter of the circumcircleis `
`      |PU| = 2`
` `

Edited on November 6, 2007, 8:07 pm
 Posted by Bractals on 2007-11-06 12:37:20

 Search: Search body:
Forums (0)