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    question on properties of circle 9

    Referring to the attached pic, find the radii of both circles.
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    Hello ukorov
    Quote Originally Posted by ukorov View Post
    Referring to the attached pic, find the radii of both circles.
    In the attached diagram, I have marked the points of contact with the smaller circle as E and F.

    Then, if the radius of the smaller circle is x
    O_1D=O_1F=DQ=FQ = x
    and if PE = y, then PD = y (tangents from a point to a circle are equal)

    PQ = x+y=6

    In \triangle TPQ, TP= 10 cm (Pythagoras)

    TE = TF \Rightarrow 10-y = 8-x \Rightarrow y-x=2

    Solving these simultaneous equations: y = 4, x = 2.

    So the radius of the smaller circle is 2 cm.

    Similarly let PA = PC = w and QC = QB = z

    Express the lengths of TA and TB in terms of w and z. Then use the fact that PQ = 6 to form another equation. Solve these equations to find z, the radius of the larger circle.

    Grandad
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    Quote Originally Posted by Grandad View Post
    Hello ukorovIn the attached diagram, I have marked the points of contact with the smaller circle as E and F.

    Then, if the radius of the smaller circle is x[/size]
    O_1D=O_1F=DQ=FQ = x
    and if PE = y, then PD = y (tangents from a point to a circle are equal)

    PQ = x+y=6

    In \triangle TPQ, TP= 10 cm (Pythagoras)

    TE = TF \Rightarrow 10-y = 8-x \Rightarrow y-x=2

    Solving these simultaneous equations: y = 4, x = 2.

    So the radius of the smaller circle is 2 cm.

    Similarly let PA = PC = w and QC = QB = z

    Express the lengths of TA and TB in terms of w and z. Then use the fact that PQ = 6 to form another equation. Solve these equations to find z, the radius of the larger circle.

    Grandad
    PQ = 6 = w + z
    w = 6 - z .....(1)
    TA = TP + w = 10 + w
    TB = TQ + z = 8 + z
    hence 10 + w = 8 + z .....(2)
    (1) to (2):
    10 + 6 - z - 8 = z
    2z = 8
    z = 4
    hence larger circle has radius 4cm.

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