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Help with geometry problem?

http://www.math.uwo.ca/~mdawes/woma/problems/potw9...

Given: Larger pentagon is regular.

Prove that the pentagon created by the inscribed star is regular. Also, find the ratio between the area of the smaller pentagon to the larger pentagon.

1 Answer

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  • Andy
    Lv 5
    1 decade ago
    Favorite Answer

    Label the vertices of the outer pentagon, clockwise, starting at the top, A, B, C, D, E.

    Label the vertices of the inner pentagon, clockwise, starting at the bottom, F,G,H,I,J

    Since ABCDE is regular, then ADE is similar to ABC by SAS, thus AD and AC are

    equal in length. By the same technique, BD and BD are equal, AC and CE are equal,

    and AD and BD are equal. Thus AC = AD = BC = BE = CE.

    Since ABCDE is regular, then BCD is similar to ABC by SSS. By the same technique,

    all the triangles ABC, BCD, CDE, DEA, and EAD are similar. Thus all ten angles

    CAB, ABE, DBC, ACB, ECD, BDC, ADE, CED, AEB, and DAE have the same angle

    measure.

    Since angles CAB and ABE have the same angle measure as angles DAE and AEB,

    and since AE = AB, triangles AIB and AHE are similar by ASA. By the same technique,

    all five triangles, AIB, AHE, EGD, DFC, and CJB are similar.

    Since the triangles AHB, AIB, and BJC are similar, then EH = IB = AI = JC, and since

    EB = AC, we have HI = IJ. Similarly, IJ = JF = FG = GH, so FGHIJ is regular.

    My personal view is that a question should be a single question, not two or more.

    My guess is that others feel the same, which is why mine is the only answer to date.

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