What's the ratio between the segments A F . B G F G in the figure below?

Question

What&#039;s the ratio between the segments             A      F      .      B      G              F      G       in the figure below?

Lea Johnson · Accepted Answer

Step 1Notice that points H and I in this answer are different from those in the original post.Let             A      F      ⋅      B      G              F      G        =  R . First of all,  A  B  =  A  F  +  F  G  +  B  G  =  2            15      2        −          13      2        =  4      14  Power of a point gives us                    A        F        ⋅        F        B                            =        A        F        ⋅                  (          4                      14                    −          A          F          )                =        E        F        ⋅        D        F                                  (        1        )                            B        G        ⋅        A        G                            =        B        G        ⋅                  (          4                      14                    −          B          G          )                =        E        G        ⋅        G        C                                  (        2        )            and it follows that                                          (            1            )            ⋅            (            2            )                                F                          G              2                                              ⟹                R                  (          4                      14                    +          R          )                =                              E            F                                F            G                          ⋅                              E            G                                F            G                          ⋅        D        F        ⋅        G        C            Now, with   △  E  F  G  ∼  △  H  D  F we have             E      F              F      G        =            H      D              D      F       . Similarly, with   △  E  F  G  ∼  △  I  G  C we have             E      G              F      G        =            I      C              G      C      . Therefore,  R      (    4          14        +    R    )    =  H  D  ⋅  I  CSince   △  H  D  F  ∼  △  I  G  C , it follows that  H  D  ⋅  I  C  =  H  F  ⋅  I  G  =      13    2    =  169and we finally have the quadratic equation      R    2    +  4      14    R  −  169  =  0whose solutions are  R  =  −  2      14    ±  15We&#039;ll take       R    =    −    2          14        +    15   since   R  &amp;gt;  0 .Maybe there are solutions that are more straightforward or concise, but I&#039;m just gonna put my answer out here. I believe   R  =  −  2      14    +  15 is the correct answer since I&#039;ve verified it on GeoGebra. Some lengthy brute-force simplifications are skipped in this answer.

dumnealorjavgj · Accepted Answer

Step 1That leads to                         B        C                    A        F              =                  C        E                    A        E                   ,                  A        D                    B        G              =                  A        E                    G        E            Multiplying,                     (1)                                                        B              C              ×              A              D                                      A              F              ×              B              G                                =                                    C              E                                      G              E                                          As       △    F    E    G    ∼    △    D    E    C    ,                       C        E                    G        E              =                  C        D                    F        G            Plugging into (1),                    A        F        ×        B        G                    F        G              =                  B        C        ×        A        D                    C        D            As it is a cyclic trapezium, it must be isosceles and   B  C  =  A  D . Applying Pythagoras,  O      H    2    =  O      B    2    −  B      H    2    =      15    2    −      13    2    =  56So,   C  H  =  15  −      56    B  C  ×  A  D  =  B      C    2    =  B      H    2    +  C      H    2    =      13    2    +  (  15  −      56        )    2    =  30  (  15  −  2      14    )So the answer is      (  15  −  2      14    )

What's the ratio between the segments A F . B G <

Answered question

Answer & Explanation

New Questions in Elementary geometry