I am given 2 side lengths for one triangle and two side lengths for the parallelogram. I am asked to find the length of m (FE) and n (DE)given the lenghts:h (AC) = 9k (AF) = 15f (AB) = 16I don't see how to use Law of Sines because I don't have any angles and I don't see how to use Law of Cosines to solve triangle ACF because I am missing a side length.

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I am given 2 side lengths for one triangle and two side lengths for the parallelogram. I am asked to find the length of m (FE) and n (DE)given the lenghts:h (AC) = 9k (AF) = 15f (AB) = 16I don&#039;t see how to use Law of Sines because I don&#039;t have any angles and I don&#039;t see how to use Law of Cosines to solve triangle ACF because I am missing a side length.

zalitiaf · Accepted Answer

Let&#039;s let  A  =  (  0  ,  0  )  B  =  (  16  ,  0  )  C  =  (  9  cos  ⁡  θ  ,  9  sin  ⁡  θ  )   for some   0  &amp;lt;  θ  &amp;lt;      90      ∘    D  =  B  +  C  =  (  16  +  9  cos  ⁡  θ  ,  9  sin  ⁡  θ  )  F  =      (                  15        2            −              9        2                    sin        2            ⁡      θ        ,    9    sin    ⁡    θ    )  Then we have a parallelogram   A  B  D  C with a point   F on   C  D, such that   |  A  C  |  =  9,   |  A  F  |  =  15, and   |  A  B  |  =  16. Our next step is to find   E. We&#039;ll be done if we can show that   |  A  E  | is a non-constant function of   θ.  E  is the intersection of the lines determined by segments   A  F and   B  D. To find the coordinates of   E, we&#039;ll first find the equations of these lines. Using the point-slope form, we have that the equation for the line determined by segment   A  F is:  y  =            9      sin      ⁡      θ                      15        2            −              9        2                    sin        2            ⁡      θ        ⋅  xUsing the point-slope form, we have that the equation for the line determined by segment   B  D is:  y  =            9      sin      ⁡      θ              9      cos      ⁡      θ        ⋅  (  x  −  16  )Hence we can find the   x-coordinate of   E by solving            9      sin      ⁡      θ                      15        2            −              9        2                    sin        2            ⁡      θ        ⋅  x  =            9      sin      ⁡      θ              9      cos      ⁡      θ        ⋅  (  x  −  16  )This gives us that  x  =            16      ⋅                        15          2                −                  9          2                          sin          2                ⁡        θ                                      15          2                −                  9          2                          sin          2                ⁡        θ            −      9      cos      ⁡      θ      We can then plug this into the equation for the line determined by segment   A  F to obtain that  y  =            16      ⋅      9      sin      ⁡      θ                                15          2                −                  9          2                          sin          2                ⁡        θ            −      9      cos      ⁡      θ      Hence  E  =      (                  16        ⋅                              15            2                    −                      9            2                                sin            2                    ⁡          θ                                                  15            2                    −                      9            2                                sin            2                    ⁡          θ                −        9        cos        ⁡        θ              ,                  16        ⋅        9        sin        ⁡        θ                                          15            2                    −                      9            2                                sin            2                    ⁡          θ                −        9        cos        ⁡        θ              )  It follows that  |  A  E  |  =            16      ⋅      15                                15          2                −                  9          2                          sin          2                ⁡        θ            −      9      cos      ⁡      θ      Note that if   θ  =      30      ∘  , then   |  A  E  |  ≈  36.8, but if   θ  =      60      ∘  , then   |  A  E  |  ≈  28.9. So   |  A  E  | is a non-constant function of   θ.Finally, note that   m  =  |  A  E  |  −  15. So   m is a non-constant function of   θ. We need more information.

I am given 2 side lengths for one triangle and two side lengths for the parallelogram. I am asked to

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