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Math Help - High School Math Problem Involving Circles, Radii and Tangent (Image Included)

  1. #1
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    High School Math Problem Involving Circles, Radii and Tangent (Image Included)



    I've been trying to solve this problem for a while now, but I still cannot do it. I understand that AB has a length of 5cm, and so if BC is "x" then AC = 5 + x. From there, I'm stuck. Please can somebody show me how to do it. Thanks!
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    Member Danshader's Avatar
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    well first you need to find the angle at which the line AB makes with the horizontal (angle of ABX)

    from the picture i attach, angle of ABX can be found by:
    tan ABX = (AE - BD) / ED
    tan ABX = (radius of larger circle - radius of smaller circle)/(radius of larger circle + radius of smaller circle)
    tan ABX = (4-1)/4+1)
    tan ABX = 3/5 = 0.6
    ABX = 30.9637 degrees

    from observation this will also be the angle of ACE:

    sin ACE = AE / AC
    sin 30.9637 = (radius of larger circle) / AC
    AC = radius of larger circle/sin 30.9637
    AC = 7.774603526 cm
    Attached Thumbnails Attached Thumbnails High School Math Problem Involving Circles, Radii and Tangent (Image Included)-asdasdasd.jpg  
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  3. #3
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    Thanks! But why are the the two angles equal?
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    Member Danshader's Avatar
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    imagine moving the line BX down perpendicularly to the vertical until it intercepts line EDC(above line EDC). Since line AC is straight and line BX is parallel to line EDC the angles will also be the same.
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    Hmm. I've got the mark scheme here, and it has a different answer.

    I couldn't quite understand the working, that's why I came here.

    It says:

    BC/1 = (BC + 5)/4
    4x = x + 5
    x = (5/3)
    AC = 6 + (2/3)
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    Member Danshader's Avatar
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    ahhh i see my mistake now sorrie >.<

    it is suppose to be

    sin ABX = (radius of larger circle - radius of smaller circle)/(radius of larger circle +radius of smaller circle)
    sin ABX = 4-1/4+1 = 0.6

    again using observation angle ACE = angle ABX

    sin ACE = radius of larger circle/AC
    0.6 = 4/AC
    AC = 4/0.6
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  7. #7
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    But doesn't AB = XB = ED? So why don't you get the same angle?
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    Member Danshader's Avatar
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    lol thats where i made my mistake. i mistakenly assumed that AB = ED however it is not actually equal because the circle is not drawn up to scale. the smaller circle is actually very much smaller.


    from the attachment AY is not equal in length to EZ. The only possibility for AB = ED is when both circles are equal in size(same radius)
    Attached Thumbnails Attached Thumbnails High School Math Problem Involving Circles, Radii and Tangent (Image Included)-asdasdasd.jpg  
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  9. #9
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    Ok, that makes sense now.

    But, I still don't understand the mark scheme working:
    How did they get:

    BC/1 = (BC + 5)/4

    and

    4x = x + 5
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  10. #10
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    Also, surely AB has to be larger than XB, as it is the hypotenuse. By looking at the picture, BX's length is the two radii added together, plus a bigger gap. I'm confused.
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  11. #11
    Super Member wingless's Avatar
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    BDC and AEC are similar.

    So,
    \frac{|BC|}{|AC|} = \frac{|BD|}{|AE|}

    \frac{1+x}{6+x}=\frac{1}{4}

    x = \frac{2}{3}

    |AC| = 6 + x = 6 + \frac{2}{3} = \frac{20}{3}
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  12. #12
    Member Danshader's Avatar
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    well they are using the same method but more vaguely.

    consider the triangle BCD and triangle ACE. obviously we can see that triangle BCD is the smaller version of triangle ACE.

    from the attachment,
    for the smaller triangle,
    cos theta = BD/BC ====(1)

    for the larger triangle,
    cos theta = AE/(AB+BC) ======(2)

    from (1) and (2)

    BD/BC = AE/(AB+BC)
    BC/BD = (AB+BC)/AE

    let BC = x
    x/1 = ((4+1)+x)/4
    4x = 5+x
    3x = 5
    x=5/3 = BC

    now AC = AB + BC = 4+1+5/3
    =5+ 3/3 + 2/3
    =6 + 2/3
    Attached Thumbnails Attached Thumbnails High School Math Problem Involving Circles, Radii and Tangent (Image Included)-asdasdasd.jpg  
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  13. #13
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    Do you mean Sin Theta, not Cos Theta?

    Ok, I understand how to work out the answer now using the similar triangles method. But, using the trig method, I can't understand how AB is bigger than BX (the horizontal).
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    Member Danshader's Avatar
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    well you already got the thinking right. the triangle ABX has a hypotenuse of AB and hypotenuse are always longer than any other side length of the triangle.

    For exampleattachment)
    sin theta = 8/10 = 0.8 ===> means that the line OB is a fraction of line AB
    sin alpha = 6/10 = 0.6 =====> means that line OA is a fraction of line AB
    this is true for all triangles

    in our case A=A, B=B,O=X.
    cos ABX = BX/AB ====> line BX is a fraction of length of line AB(BX<AB)
    Attached Thumbnails Attached Thumbnails High School Math Problem Involving Circles, Radii and Tangent (Image Included)-asdasdasd.jpg  
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  15. #15
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    I'm still confused. I appreciate you for all trying to help me, and I can understand all the methods of working this out.

    But I still don't understand why AB is larger then BX (the horizontal).

    I can't understand your diagram that well. D:
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