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Math Help - circle (2)

  1. #1
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    circle (2)

    In the figure , ST is the common tangent at P to the two circles . PVW and PQR are two straight lines . OS bisects \angle PSR and PV=PS , the lines QV and RW are also parallel .

    Prove that SR=VW .
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  2. #2
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    Hello thereddevils
    Quote Originally Posted by thereddevils View Post
    In the figure , ST is the common tangent at P to the two circles . PVW and PQR are two straight lines . OS bisects \angle PSR and PV=PS , the lines QV and RW are also parallel .

    Prove that SR=VW .
    I've re-drawn the diagram (attached), so that QS (which I think is what you meant) looks more like the bisector of \angle PSR.

    Then in \triangle's SQP, SQR
    SP = SR (tangents from a point to a circle)

    \angle PSQ = \angle SRQ ( SQ is angle bisector, given)

    QS is common
    \therefore \triangle's SQP, SQR are congruent (SAS)

    \therefore PQ=RQ

    Now - and I'll leave you to fill in the details - prove that \triangle's PVQ, PWR are similar. Hence PV:VW=PQ:QR, and hence PV=VW.

    But PV = PS (given). Therefore ... ?

    Grandad
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    Quote Originally Posted by Grandad View Post
    Hello thereddevilsI've re-drawn the diagram (attached), so that QS (which I think is what you meant) looks more like the bisector of \angle PSR.

    Then in \triangle's SQP, SQR
    SP = SR (tangents from a point to a circle)

    \angle PSQ = \angle SRQ ( SQ is angle bisector, given)

    QS is common
    \therefore \triangle's SQP, SQR are congruent (SAS)

    \therefore PQ=RQ

    Now - and I'll leave you to fill in the details - prove that \triangle's PVQ, PWR are similar. Hence PV:VW=PQ:QR, and hence PV=VW.

    But PV = PS (given). Therefore ... ?

    Grandad
    Thanks again Grandad , what software do you use to produce such a neat diagram ?
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    Hello thereddevils
    Quote Originally Posted by thereddevils View Post
    Thanks again Grandad , what software do you use to produce such a neat diagram ?
    I have CorelDraw version 12, and Corel Photo-paint 12 on my maching, but they're much too heavyweight for this. So I just use the drawing tools in Word, which are very quick and easy to use. Any text I put in a text box, with no border or background colour. Then, with the image on screen, I use a screen-capture program (MWSnap - freeware, donations appreciated; works fine in Vista) to capture the image to clipboard; paste into Paint; save and then crop in Windows Photo Gallery. Sounds complicated, but it's actually very quick. Graphs I tend to draw in Excel, copy, paste, crop and upload.

    Grandad
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    circle (2)

    originally posted by thereddevils

    Hello reddevil and grandpa'

    The diagrames so far do not appear correct. The initial data are not adequate

    Try this
    O is the center of the larger circle.The smaller circle passes thru 0
    When SP=PV the diagram is unique (only one)Other diagrams SP is not equal PV but always PV=VW andPQ=QR


    bjh
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