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

 Convex Quadrilateral And Intersection (Posted on 2007-10-03)
PQRS is a convex quadrilateral with diagonals PR and QS that intersect at the point T. It is known that, Angle QPR = 50o, Angle RPS = 60o, Angle RQS = 30o and Angle QSR = 25o

Determine Angle PTQ.

 See The Solution Submitted by K Sengupta No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution | Comment 4 of 8 |
`Let a, b, and x be the measures in degrees ofangles RST, RQT, and PTQ respectively. Applyingthe sine rule to the four triangles sharingvertex T we get`
`    |PT|          |PT|         |QT|        |QT| ----------- = ---------- = ---------- = ---------  sin(x+2a)     sin(PQT)     sin(QPT)     sin(2a)`
`    |QT|         |QT|         |RT|        |RT| ---------- = ---------- = ---------- = --------  sin(x-b)     sin(QRT)     sin(RQT)     sin(b)`
`   |RT|        |RT|         |ST|         |ST| -------- = ---------- = ---------- = ---------  sin(a)     sin(RST)     sin(SRT)     sin(x+a)`
`   |ST|         |ST|         |PT|         |PT| --------- = ---------- = ---------- = -----------  sin(2b)     sin(SPT)     sin(PST)     sin(x-2b)`
`Eliminating the lengths from the above fourequations we get`
`  sin(x-b)   sin(x+2a)     sin(x+a)   sin(x-2b) ---------- ----------- = ---------- -----------   sin(b)    sin(2a)        sin(a)     sin(2b)`
` [cot(b)sin(x)-cos(x)][cot(2a)sin(x)+cos(x)]   = [cot(a)sin(x)+cos(x)][cot(2b)sin(x)-cos(x)]`
`     cot(2a)cot(b)sin(x)^2    + [cot(b)-cot(2a)]cos(x)sin(x)   - cos(x)^2  =   cot(a)cot(2b)sin(x)^2   + [cot(2b)-cot(a)]cos(x)sin(x)   - cos(x)^2`
`           cot(2b) - cot(a) - cot(b) + cot(2a) tan(x) = -------------------------------------              cot(2a)cot(b) - cot(a)cot(2b)`
`Using the following identity `
`            cot(t)^2 - 1 cot(2t) = --------------              2 cot(t)`
`we get`
`           cot(a)cot(b) + 1 tan(x) = ------------------ = tan(90+[b-a])           cot(b) - cot(a)`
`Therefore,`
` x = 90 + b - a`
`For our problem,`
` x = 90 + 30 - 25 `
`   or`
` Angle PTQ = 95 degrees`
` `

 Posted by Bractals on 2007-10-03 12:26:50

 Search: Search body:
Forums (0)