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Math Help - easy geometry!

  1. #1
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    easy geometry!

    Let be a line segment in the plane and a points between and . Construct isosceles triangles and on one side of the segment such that and an isosceles triangle on the other side of such that
    show that  AP+CQ-AR = 0
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    Well, my teacher gave a solution , but as my trig is horrible , ididnt understand it.....
    now we know  angles PAB , PBC, QBC, QCB, RAC, ACR = 30
    Now  AP \sin 30 + CQ \sin 30 = \frac{1}{2} AC = AR \sin 30 , which implies  AP+CQ = AR OR, AP+CQ-AR=0
    It may be very trivial but i dont understand why  AP \sin 30 + CQ \sin 30 = \frac{1}{2} AC = AR \sin 30
    please help!
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by earthboy View Post
    Let be a line segment in the plane and a points between and . Construct isosceles triangles and on one side of the segment such that and an isosceles triangle on the other side of such that
    show that  AP+CQ-AR = 0
    Note that \displaystyle \triangle APB and \displaystyle \triangle BQC are both isosceles triangles with the same side lengths. Similar triangles with exactly the same magnitude. (do you see why?)

    B bisects AC (why?), and so, the bases of triangles \displaystyle APB and \displaystyle BQC are each half the length of \displaystyle \triangle ARC, which is also a similar triangle. Since these are similar, corresponding sides are proportional, and knowing \displaystyle \triangle ARC is double the magnitude of the two other triangles, we know the exact proportion. The upshot of all this is:

    \displaystyle AP + CQ - AR = \frac 12AR + \frac 12 AR - AR = 0

    since AP and CQ are corresponding sides to AR with half its magnitude.


    Now I spouted a lot of unproven claims here. It's up to you to fill in the blanks
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by earthboy View Post
    Well, my teacher gave a solution , but as my trig is horrible , ididnt understand it.....
    now we know  \angle PAB , PBC, QBC, QCB, RAC, ACR = 30
    Now  AP \sin 30 + CQ \sin 30 = \frac{1}{2} AC = AR \sin 30 , which implies  AP+CQ = AR OR, AP+CQ-AR=0
    It may be very trivial but i dont understand why  AP \sin 30 + CQ \sin 30 = \frac{1}{2} AC = AR \sin 30
    please help!
    Fine, so we want to use trig. Very well. We know the magnitude of all the angles because we see that we had similar triangles here, they are all isosceles, and hence the other angles were 30 degrees each.

    construct two perpendicular lines to AC, one passing through P, the other through Q. That will give you the sine relationships your professor mentioned (Remember SOHCAHTOA!) The fact that the sum is 1/2 AC goes back to what I was saying about similar triangles with the factor of 1/2
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    Hi earthboy,
    Is B supposed to be the midpoint of AC?

    bjh
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    Quote Originally Posted by bjhopper
    Is B supposed to be the midpoint of AC
    the question says, B is any point on the line AC.
    but then
    Quote Originally Posted by Jhevon
    B bisects AC (why?)
    omg!its becoming more confusing for me is the solution even correct ?
    sorry if i am being silly :-o.
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  7. #7
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    Hi earthboy,
    With B as any point on a line segment AC the statement AP +CQ - AR =0 can be proved by easy geometry (no trig required ).Other info as you described.


    bjh
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  8. #8
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by earthboy View Post
    the question says, B is any point on the line AC.
    but then omg!its becoming more confusing for me is the solution even correct ?
    sorry if i am being silly :-o.
    the problem says B is a point on AC, not "any" point. it ends up being the midpoint with the way the rest of the problem is set up. the secret that tells us this lies in the angles APB and BQC....
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  9. #9
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    easy geometry problem

    Hello jhevon,
    A point on a line can be placed anywhere if there is no spec given.To draw a diagram for a proof lets make AB= 4 cm BC = 6 cm AC =10cm. Points P, Q, R lie on the perpendicular bisectors of AB,BC, AC respectively. Three 30-30-120 isosceles triangles are erected as described. Extend AP and QC to meet the perpendicular bisector of AC at S above AC

    ARCS and PBQS are parallelograms AP+QC = AR.The whole is the sum of the parts
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    yes ...
    sorry if i am being a moron ,but ....
    Quote Originally Posted by bjhopper
    Extend AP and QC to meet the perpendicular bisector of AC at S above AC
    suppose AP and QC when extended meet as S. Can we be sure that the perpendicular bisector of AC always passes through the point S ???

    another doubt: is the original trig solution ok?

    HELP!
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  11. #11
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    Hi earthboy,
    Did you draw the diagram ? Triangle ACS is an isosceles triangle congruent to ARC.AP and CQ extended meet at S on the perpendicular bisector.I cannot believe you received the trig relations from a math teacher.

    bjh
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